Download Spectral Properties of Schrödinger Operators

Universität Stuttgart
Fachrichtung Mathematik
Institut für Analysis Dynamik und Modellierung
Spectral Properties
of Schrödinger Operators
Habilitationsschrift
Hynek Kovařı́k
2008
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Mým rodičům Marii a Ivanovi.
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Acknowledgements
Many people have given me a lot of support in recent years. I would like to use
this occasion to thank
- Timo Weidl, under whose supervision I worked at Stuttgart University.
I could benefit from his knowledge during numerous discussions, which
helped me improve my understanding of mathematics considerably.
- Pavel Exner, my PhD advisor, who introduced me to the field of mathematical physics, and never stopped to care about my scientific development.
- my friend Tomas Ekholm, with whom I had the pleasure to work on several
projects in the past and with whom I hopefully will also work in the future.
- my co-authors Denis Borisov, Tomas Ekholm, Pavel Exner, Rupert Frank,
David Krejčiřı́k, Semjon Vugalter and Timo Weidl.
- Steffi Siegert for proof reading of the German introduction.
- my fiancée Riccarda Rossi. I would never have been able to finish this
thesis without her love and encouragement.
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Contents
Deutsche Zusammenfassung
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1 Introduction
1.1 Preliminaries . . . . . . . . . . . . . . .
1.2 Schrödinger operators in waveguides . .
Hardy inequalities for Laplace operators
Waveguides with magnetic field . . . . .
Twisted waveguides . . . . . . . . . . .
1.3 Spectral estimates . . . . . . . . . . . .
Discrete spectrum in Rn . . . . . . . . .
Metric trees . . . . . . . . . . . . . . . .
Weak coupling behaviour . . . . . . . .
Weighted Lieb-Thirring inequalities . . .
1.4 Two-dimensional Schrödinger operators
Logarithmic Lieb-Thirring inequalities .
Applications to quantum layers . . . . .
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15
16
17
18
22
24
28
28
32
33
34
36
38
39
2 Schrödinger operators in waveguides
47
2.1 Waveguides with magnetic field . . . . . . . . . . . . . . . . . . . 47
2.2 Waveguide with magnetic field and combined boundary conditions 77
2.3 A Hardy inequality in twisted waveguides . . . . . . . . . . . . . 95
2.4 Twisted Dirichlet-Neumann waveguide . . . . . . . . . . . . . . . 119
2.5 Periodically twisted tube . . . . . . . . . . . . . . . . . . . . . . . 131
3 Metric trees
143
3.1 Weak coupling behaviour . . . . . . . . . . . . . . . . . . . . . . 143
3.2 Weighted Lieb-Thirring inequalities . . . . . . . . . . . . . . . . . 163
4 Spectral estimates in two dimensions
197
4.1 Logarithmic Lieb-Thirring inequality . . . . . . . . . . . . . . . . 197
4.2 Applications to quantum layers . . . . . . . . . . . . . . . . . . . 211
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Deutsche Zusammenfassung
Diese Arbeit enthält eine Sammlung der wichtigsten Resultate meiner Forschungsarbeiten an der Universität Stuttgart. Das Ziel dieser Zusammenfassung ist es,
dem Leser einen kurzen Überblick über die erzielten Ergebnisse zu verschaffen.
Das vorgestellte Material wird thematisch in drei Kapitel verteilt.
Kapitel 2: Schrödinger-Operatoren in Wellenleitern.
Unter Quantenwellenleitern verstehen wir entweder Streifen (zwei-dimensionaler
Fall) oder Röhren (drei-dimensionaler Fall) mit sehr geringer Breite von einer
Ordnung von 10 bis 100 Nanometern. Diese Strukturen werden für die Untersuchung einiger wichtiger Elemente der Nanoelektronik als mathematische Modelle benutzt. Die Bewegung der Teilchen in solchen Wellenleitern wird durch
spektrale Eigenschaften der entsprechenden Schrödinger-Gleichung beschrieben.
Die dazugehörige mathematische Aufgabe besteht aus der Spektralanalysis von
Differentialoperatoren, in diesem Fall Laplace- oder Schrödinger-Operatoren, im
gegebenen Gebiet.
Der Zusammenhang zwischen spektralen Eigenschaften des Laplace-Operators
in Wellenleitern und der Geometrie der Wellenleitern ist seit längerer Zeit bekannt,
[12, 14, 17, 21, 22, 31, 34, 39, 44, 68]. Insbesondere ist bekannt, siehe [21, 34],
dass durch geeignete Krümmung eines Wellenleiters gebundene Zustände entstehen. Diese Zustände entsprechen den im Wellenleiter lokalisierten Teilchen.
Solche Effekte werden mathematisch durch diskrete Eigenwerte des Operators
−∆Ω
in L2 (Ω)
beschrieben, wobei Ω einen Streifen bzw. eine Röhre darstellt. Wir bezeichnen
mit Ω0 = R × ω den geraden Wellenleiter mit dem Querschnitt ω ∈ Rn−1 und
mit Ω einen Wellenleiter, der durch verschiedene geometrische Deformationen
(Krümmung, Verdrehung usw.) von Ω0 entsteht.
Der Operator −∆Ω ist der Dirichlet-Laplace-Operator, der auf dem Rand
von Ω die Dirichlet-Randbedingungen erfüllt. Ist die Krümmung von Ω lokal,
so stimmt das wesentliche Spektrum von −∆Ω mit der Halbgeraden [λ1 , ∞)
überein, wobei λ1 den kleinsten Eigenwert des Dirichlet-Laplace-Operators auf
dem Querschnitt ω des Wellenleiters bezeichnet. Die gebundenen Zustände
entsprechen dann den Eigenwerten von −∆Ω , die unterhalb von λ1 liegen.
Die Arbeiten [21, 34, 39] zeigen, dass jede lokale Krümmung, egal wie klein,
mindestens einen solchen Eigenwert produziert und so zur Lokalisierung des
Teilchens im Wellenleiter führt. Ähnliche Effekte entstehen durch lokale Ausbreitung des Wellenleiters, siehe [14]. Dasselbe gilt natürlich auch für Störungen,
die nicht von geometrischer Natur sind, z.B. für Potentialstörungen vom Typ
−∆Ω0 + V , wobei V das Potential darstellt.
Wellenleitern mit Magnetfeld. In [27], Abschnitt 2.1, haben wir gezeigt,
dass die oben beschriebenen Effekte der Krümmung, Ausbreitung und Potentialstörungen, die zur Existenz der gebundenen Zustände führen, bis zu einem
9
gewissen Grad kompensiert werden können, indem man den zwei-dimensionalen
Wellenleiter in ein Magnetfeld platziert. Dabei wird vorausgesetzt, dass das
Magnetfeld lokalisiert ist und senkrecht zur Ebene des Wellenleiters steht. Mathematisch heißt das, dass wir statt −∆Ω0 den Operator
HA = (−i∇ + A)2
in L2 (Ω0 )
(1)
betrachten, wobei B = rot A den Vektor des Magnetfeldes bezeichnet. Da das
Magnetfeld lokalisiert ist (kompakt getragen), bleibt das Spektrum von HA
bezüglich dem von −∆Ωo unverändert, d.h., σ(HA ) = [λ1 , ∞).
Auf der anderen Seite führt die Existenz des Magnetfeldes dazu, dass das
Spektrum von HA im folgenden Sinne stabiler wird: Die gebundenen Zustände,
d.h. die Eigenwerte von HA kleiner als λ1 , erscheinen nicht für jede Krümmung,
sondern nur dann, wenn diese Krümmung stark genug wird. Der Beweis dieses
Resultates basiert darauf, dass HA eine Ungleichung vom Typ
HA − λ1 ≥ ρA ,
ρA ≥ 0,
(2)
erfüllt, siehe Abschnitt 2.1. Eine solche Abschätzung wird Hardy-Ungleichung
genannt und die Funktion ρA , die von dem Magnetfeld abhängt, wird als HardyGewicht bezeichnet. Ungleichung (2) sagt aus, dass, um einen Eigenwert aus
dem wesentlichen Spektrum von HA herauszuziehen, d.h. das Teilchen zu binden,
es nicht ausreicht, irgendeine negative Störung zu HA zu addieren, sondern dass
eine Störung nötig ist, die im gewissen Sinne stärker als das Hardy-Gewicht
ρA ist. Diese Störung kann entweder eine Potentialstörung sein oder eine geometrische Deformation des Wellenleiters (Krümmung usw.). Dieses Ergebnis
wurde in [11], siehe Abschnitt 2.2, auf Wellenleitern mit gemischten Randbedingungen verallgemeinert.
Wellenleitern mit Verdrehung. Eine gewisse Stabilität des Spektrums des
Laplace-Operators in Ω kann auch ohne Magnetfeld erreicht werden und zwar
durch lokale Verdrehung eines drei-dimensionalen Wellenleiters, dessen Querschnitt ω nicht kreisförmig ist. Dies wurde in [28] bewiesen, siehe Abschnitt
2.3. In diesem Fall wird der Operator
−∆Ω
in L2 (Ω)
(3)
untersucht. Hier beschreibt Ω = fθ (Ω0 ) den verdrehten Wellenleiter, wobei
fθ (x1 , x2 , x3 ) = (x1 , x2 cos θ(x1 ) + x3 sin θ(x1 ), x3 cos θ(x1 ) − x2 sin θ(x1 )) ,
und θ : R → R bezeichnet den Winkel, um den der Querschnitt des Wellenleiters um seine Achse rotiert wird, siehe Bild 1.1 auf Seite 25. Der gerade
Wellenleiter (ohne Verdrehung) entspricht θ ≡ const. Sobald θ 6≡ const, werden die schwach gebundenen Zustände aufgehoben ähnlich wie in dem oben
beschriebenen Modell mit lokalem Magnetfeld. Auch in diesem Fall gilt die
entsprechende Hardy-Ungleichung
−∆Ω − λ1 ≥ ρθ̇ ,
10
ρθ̇ ≥ 0,
(4)
dθ
, siehe [28]. Dabei hat das Hardy-Gewicht ρθ̇ eine spezielle Strukwobei θ̇ = dx
1
tur: es hängt nicht nur von θ̇ ab, sondern auch von der Geometrie des Querschnitts ω, siehe Abschnitt 2.3. Ist zum Beispiel ω kreisförmig, so wird, wie
erwartet, ρθ̇ ≡ 0 egal wie groß θ̇ ist. In [47], siehe Abschnitt 2.4, wurde gezeigt,
dass für zwei-dimensionale Wellenleitern ähnliches Ergebnis erzielt wird durch
den so genannten “twist” der Randbedingungen, siehe Bild 1.2 auf Seite 27.
Wellenleitern mit periodischer Verdrehung werden in Abschnitt 2.5 betrachtet.
Schließlich machen wir darauf aufmerksam, dass ohne Magnetfeld bzw. Verdrehung kein nichttriviales Hardy-Gewicht ρ existiert, für welches die Ungleichung (2) bzw. (4) gilt.
Kapitel 3: Spektralabschätzungen auf metrischen Bäumen.
Metrische Bäume bilden eine spezielle Klasse der so genannten Quantengraphen,
nämlich der Graphen, auf denen zwei beliebige Punkte durch einen eindeutigen Weg verbunden sind. Spektralprobleme für Laplace- und SchrödingerOperatoren auf solchen Strukturen wurden in den letzten Jahren intensiv erforscht, [15, 30, 51, 48, 63, 62, 72, 73]. In den Arbeiten [46, 26] haben wir
den Zusammenhang zwischen der Geometrie solcher metrischer Bäume und den
Spektraleigenschaften des Schrödinger-Operators untersucht.
Geometrie eines metrischen Baumes. Ein metrischer Baum Γ mit der
Wurzel o besteht aus der (abzählbaren) Menge der Knoten V(Γ) und aus der
Menge der Kanten E(Γ), d.h. ein-dimensionalen Intervallen, die die Knoten
miteinander verbinden. Für gegebenes x ∈ Γ bezeichnen wir mit |x| die Länge
des (eindeutigen) Weges zwischen x und der Wurzel o. Ist z ∈ V(Γ) ein Knoten,
so definieren wir die Verzweigungszahl b(z) als die Anzahl der Kanten, die von z
ausgehen. Die Generation eines Knotens z ∈ V(Γ) wird definiert als die Anzahl
der Knoten (inklusive der Wurzel), die auf dem Weg zwischen z und der Wurzel
o liegen. Wir werden nur mit regulären Bäumen arbeiten, d.h. mit Bäumen, auf
denen alle Knoten derselben Generation die gleiche Verzweigungszahl haben,
und alle Kanten, die aus diesen Knoten ausgehen, die gleiche Länge haben.
Die globale Geometrie des Baumes Γ werden wir mittels der so genannten
Verzweigungsfunktion g0 beschreiben. Diese Funktion wird durch
g0 (t) = #{x ∈ Γ : |x| = t}
definiert. Analog zu den Euklidischen Räumen definieren wir dann die globale
Dimension, siehe [46], von Γ durch das Wachstum der Funktion g0 : Existieren
zwei positive Konstanten c1 und c2 und eine reelle Zahl d ≥ 1, so dass
c1 ≤
g0 (t)
≤ c2 ,
(1 + t)d−1
∀ t ∈ R+ ,
gilt, so nennen wir d die globale Dimension von Γ. Im Gegensatz zu den Euklidischen Räumen muss d nicht ganzzahlig sein.
11
Gewichtete Lieb-Thirring-Ungleichungen. Wir betrachten den SchrödingerOperator
−∆N − V in L2 (Γ),
(5)
wobei −∆N die Neumannschen Randbedingung in o erfüllt. Wir setzen voraus,
dass V (x) = V (|x|) ≥ 0 und bezeichnen mit Ej die negativen Eigenwerte von
−∆N − V .
Unser Ziel ist es, die aus den Euklidischen Räumen bekannten Lieb-ThirringUnglei-chungen auf metrische Bäume zu erweitern. Die Lieb-Thirring Ungleichungen, siehe [19, 43, 42, 58, 59, 54, 69, 76], liefern eine Abschätzung für
X
|Ej |γ ,
γ ≥ 0,
tr (−∆N − V )γ− =
j
mittels geeigneter Integrale des Potentials V . Es zeigt sich, [26], dass die Eigenschaften solcher Abschätzungen im Fall eines metrischen Baumes Γ sehr stark
von der globalen Dimension d von Γ abhängen.
Fall d > 2. Wenn d > 2, so haben wir die Abschätzung
Z
a+1
γ
tr (−∆N − V )− ≤ C(γ, a, Γ)
V (x)γ+ 2 (1 + |x|)a dx
(6)
Γ
für alle a ≥ 1 und alle γ ≥ (1 − a)/2, wobei C(γ, a, Γ) eine positive Konstante
ist, siehe Abschnitt 3.2. Wir beobachten, dass diese Ungleichung auch für γ = 0
gilt (falls wir a ≥ 1 wählen). In diesem Fall ergibt (6) eine obere Schranke auf
die Anzahl der negativen Eigenwerte von −∆N − V . Insbesondere folgt daraus,
dass der Operator −∆N − V für kleine Potentiale V keine negativen Eigenwerte
hat.
Fall 1 ≤ d ≤ 2. In diesem Fall besitzt der Operator −∆N − V mindestens
einen negativen Eigenwert unabhängig davon, wie klein das Potential V ist, [46].
Demzufolge gilt die Ungleichung (6) nur für ausreichend großes γ. Der minimale
Wert von γ ergibt sich durch das Parameter a und die globale Dimension d
folgendermaßen
1−a
,
2
(1 + a)(2 − d)
,
γ>
2d
1−a
γ≥
,
2
γ > 0,
γ≥
falls a ≤ d − 1 und 1 ≤ d < 2,
falls a > d − 1 und 1 ≤ d < 2,
falls
a < 1 und d = 2,
falls
a ≥ 1 und d = 2,
siehe Abschnitt 3.2. Für 1 ≤ d < 2, a = d − 1 und γ = 2−d
2 reduziert sich die
Ungleichung (6) auf
Z
2−d
V (x)(1 + |x|)d−1 dx.
tr(−∆N − V )−2 ≤ C
Γ
12
Dies entspricht genau dem asymptotischen Verhalten des kleinsten Eigenwertes
von −∆N − αV für α → 0+. In [46] wurde nämlich bewiesen, dass der Operator
−∆N − αV für ausreichend kleines (und positives) α genau einen Eigenwert
E1 (α) besitzt und dass dieser Eigenwert die asymptotische Gleichung
2
1 ≤ d < 2,
|E1 (α)| ∼ α 2−d ,
erfüllt. Dies wird in Abschnitt 3.1 genauer diskutiert.
Kapitel 4: Zwei-dimensionaler Schrödinger-Operator
Es ist wohl bekannt, dass Schrödinger-Operatoren in Dimension 2 gewisse spezielle
Eigenschaften haben. Zum Beispiel gilt die Lieb-Thirring-Ungleichung
Z
γ
γ+1
tr(−∆ − αV )− ≤ C α
V (x)γ+1 dx
(7)
R2
nur wenn γ > 0, [59]. Mit anderen Worten: das Infimum aller γ, für welche (7)
gilt, d.h. Null, wird nicht angenommen. Das ist ein prinzipieller Unterschied zu
der Situation in Dimension 1, wo die entsprechende Lieb-Thirring-Ungleichung
auch für das kleinstmögliche γ = 1/2 gilt, siehe [76]. Auf der anderen Seite
wissen wir, dass der Operator −∆ − αV in L2 (R2 ) für ausreichend kleines (und
positives) α genau einen negativen Eigenwert E1 (α) besitzt, und dass
|E1 (α)| ∼ e−4π(α
R
V )−1
,
n = 2,
(8)
siehe [71]. Daraus sieht man sofort, dass, egal wie klein γ ist, die Ungleichung
(7) für kleine Potentiale nicht das richtige Verhalten hat.
Logarithmische Lieb-Thirring-Ungleichung. Die oben genannten Nachteile
der Abschätzung (7) haben uns veranlasst, siehe Abschnitt 4.1, eine modifizierte
Lieb-Thirring-Ungleichung zu untersuchen, in der die Potenzfunktion auf der
linken Seite von (7) durch eine andere Funktion ersetzt wird. Genauer gesagt:
motiviert durch die asymptotische Gleichung (8), führen wir eine Folge Fs (·)
von Funktionen ein:

0 < t ≤ e−1 s−2 ,
 | ln ts2 |−1 ,
(9)
∀s > 0
Fs (t) :=

1,
t > e−1 s−2 .
Die Folge Fs (t) konvergiert
punktweise gegen 1 für s → ∞, und damit konP
vergiert die Summe
F
(|E
s
j |) gegen die Anzahl der negativen Eigenwerte
j
N (−∆ − αV ). Um weitere technische Begriffe nicht einführen zu müssen, formulieren wir das Ergebniss von Abschnitt 4.1 nur für sphärisch symmetrische
Potentiale. In diesem Fall existieren Konstanten C1 und C2 , so dass
X
Fs (|Ej |) ≤ C1 α kV ln(|x|/s)kL1 (B(s)) + C2 α kV kL1 (R2 )
(10)
j
13
für alle s > 0 gilt, wobei B(s) = {x ∈ R2 : |x| < s}. Die Konstanten C1 und
C2 sind von s unabhängig. Es ist leicht zu sehen, dass (10) die asymptotische
Gleichung (8) berücksichtigt.
Anwendungen für Quantenlayer. Die Quantenlayer können als zweidimensionale Modifizierung der Quantenwellenleiter angesehen werden. Ein Quantenlayer der Breite d wird typischerweise durch eine Platte Ω = R2 × (0, d)
dargestellt. In diesem Fall ist es sinnvoll mit dem verschobenem SchrödingerOperator
π2
in L2 (Ω)
(11)
H0 = −∆Ω − 2
d
zu arbeiten. Es wird angenommen, dass −∆Ω die Dirichlet-Randbedingungen
am Rand von Ω erfüllt. Das Spektrum von H0 ist dann rein stetig und überdeckt
die Halbge-rade [0, ∞). Ähnlich wie im Falle des Quantenwellenleiters entstehen
gebundene Zustände, d.h. negative Eigenwerte, durch geeignete geometrische
Deformationen von Ω, [12, 22], oder einfach durch Addieren eines negativen
Potentials V zum Operator H0 . Für ein gegebenes Potential V : Ω → R+
bezeichnen wir mit Ej die negativen Eigenwerte von
Hα = H 0 − α V .
(12)
Außerdem definieren wir
2
Ṽ (x1 , x2 ) =
d
Zd
V (x1 , x2 , x3 ) sin2
0
πx 3
d
dx3 ,
wobei x = (x1 , x2 , x3 ). Für Potentiale mit der Eigenschaft Ṽ (x1 , x2 ) = Ṽ (r),
wobei r2 = x21 + x22 , wenden wir die Ungleichung (10) an um zu zeigen, dass
X
Fs (|Ej |) ≤ c α kṼ ln r/skL1 (B(s)) + α kṼ kL1 (R2 ) + α3/2 kV 3/2 kL1 (Ω) .
j
(13)
In Abschnitt 4.1 wurde auch der Fall eines nicht symmetrischen Potentials betrachtet. Wir machen darauf aufmerksam, dass diese Abschätzung für α → 0
als auch für α → ∞ das richtige Verhalten hat. Falls α sehr klein ist, so ist die
rechte Seite von (13) dominiert durch den Term, der linear in α ist. Anderseits,
falls α sehr groß ist, so “gewinnt” der Term mit α3/2 . In der Arbeit [49], Abschnitt 4.2, haben wir ähnliche Abschätzungen auch für geometrisch induzierte
Eigenwerte bewiesen.
14
Chapter 1
Introduction
This thesis includes a collection of research papers, which deal with spectral
properties of Schrödinger operators in various quantum mechanical models. The
material is divided into three chapters, respectively corresponding to the analysis
of Schrödinger operators in waveguide-type domains and on metric trees, and
to spectral estimates for Schrödinger operators in two spatial dimensions. The
articles included are the following
• Section 2.1
T. Ekholm and H. Kovařı́k, Stability of the magnetic Schrödinger operator
in a waveguide, Comm. Partial Differential Equations 30 (2005) 539–565.
• Section 2.2
D. Borisov, T. Ekholm, and H. Kovařı́k, Spectrum of the magnetic Schrödinger
operator in a waveguide with combined boundary conditions, Ann. Henri
Poincaré 6 (2005) 327–342.
• Section 2.3
T. Ekholm, H. Kovařı́k, and D. Krejčiřı́k, A Hardy inequality in twisted
waveguides, Arch. Rational Mech. Anal. 188 (2008) 245-264.
• Section 2.4
H. Kovařı́k and D. Krejčiřı́k: A Hardy inequality in a twisted DirichletNeumann waveguide. Preprint: arXiv: math-ph/0603076. To appear in
Math. Nachr.
• Section 2.5
P. Exner and H. Kovařı́k: Spectrum of the Schrödinger operator in a
perturbed periodically twisted tube, Lett. Math. Phys. 73 (2005) 183–
192.
15
• Section 3.1
H. Kovařı́k: Weakly coupled Schrödinger operators on regular metric trees.
SIAM J. Math. Anal. 39 (2007) 1135–1149.
• Section 3.2
T. Ekholm, R.L. Frank and H. Kovařı́k: Eigenvalue estimetas for Schrödinger
operators on metric trees. Preprint: www.mathematik/uni-stuttgart.de/preprints:
2007/004.
• Section 4.1
H. Kovařı́k, S. Vugalter and T. Weidl: Spectral estimates for two-dimensional
Schrödinger operators with application to quantum layers, Comm. Math. Phys.
275 (2007) 827–838.
• Section 4.2
H. Kovařı́k and S. Vugalter: Estimates on trapped modes in deformed
quantum layers. Preprint: arXiv: 0710.1252. To apper in J. Math. Anal. Appl.
The aim of this introduction is to give a summary of the results obtained in
the above papers without elaborating on technical details, and to explain their
connections to known results in the field.
1.1
Preliminaries
According to the postulates of quantum physics, see e.g. [5], a state of a quantum
mechanical system is described by a vector ψ of a complex separable Hilbert
space H. It is usually supposed, without loss of generality, that ψ has unit
length. If the state of a system at time t = 0 is represented by a vector ψ0 , then
at any time t > 0 the system is represented by the vector
ψ(t) = e−iHt ψ0 ,
(1.1)
where H is the self-adjoint (time independent) energy operator in the Hilbert
space H1 . If ψ(t) is in the definition domain of H, then it satisfies the Schrödinger
equation
dψ
i
(t) = Hψ(t) .
(1.2)
dt
Assume that we want to describe the state of a particle with the mass m =
1/2 confined in some open domain Ω ⊂ Rn (bounded or unbounded) and moving
under the force generated by a potential function V : Ω → R. In this case
the Hilbert space H is isomorphic to L2 (Ω) and the energy operator H is the
differential operator
−∆ + V
(1.3)
1 We
use the system of units where the Planck constant ~ = 1.
16
with the Dirichlet boundary conditions at ∂Ω. Starting from an initial state
ψ0 (x), the state ψ(t, x) given by (1.1) has the following physical meaning:
|ψ(t, x)|2 represents the density of the probability of the particle to be located
at time t at a given point x ∈ Ω. Of a particular importance are the states
described by the eigenvectors of the energy operator H. Namely, if ψ0 is an
eigenvector of H with an eigenvalue E ∈ R, i.e.
H ψ0 = E ψ0 ,
(1.4)
then it follows from (1.1) that ψ(t, x) = e−iEt ψ0 (x). Since ψ(t, x) differs from
ψ0 (x) just by a phase factor, it describes the same state of the particle. In other
words, if the state of the particle is represented by the eigenvector of H, then
it is time-independent. In particular, the probability
Z
P (X) =
|ψ(t, x)|2 dx,
(1.5)
X
of finding the particle in some subset X ⊂ Ω is then constant in t. We therefore
say that the particle in such a state is localised.
It is thus important to know whether there exist real numbers E for which
(1.4) holds true with some ψ0 and, if so, how many of such numbers there are,
how large they are etc. This brings us to the spectral analysis of the operators
of type (1.3), which are referred to as Schrödinger operators. By the general
theory of self-adjoint operators in Hilbert spaces, see e.g. [10], the spectrum
σ(H) of H can be written as a disjoint union
σ(H) = σess (H) ∪ σd (H),
(1.6)
where σd (H) denotes the discrete spectrum of H, i.e. the set of isolated eigenvalues with finite multiplicity, the so-called bound states. The essential spectrum
σess (H) consists of the continuous spectrum, of the isolated eigenvalues of infinite multiplicity, and of the accumulation points of the eigenvalues of H. The
main object of our interest is the discrete spectrum σd (H) and its dependence
on the potential V and on the geometry of Ω.
1.2
Schrödinger operators in waveguides
In Chapter 2 we deal with different quantum waveguides which are in mathematical physics usually modelled by tubular semi-infinite domains in Rn . An
ideal straight waveguide is represented by Ω0 = R × ω, where the cross-section
ω is an open bounded connected subset of Rn−1 . A general waveguide Ω might
be considered as a result of various deformations (twisting, bending etc.) of the
straight waveguide Ω0 .
The most characteristic feature of these waveguides is their very small thickness. Consequently, quantum effects play an important role in the description of
such devices. As indicated in the previous section, the motion of free particles
17
confined in such domains is governed by the associated Schrödinger equation
(1.2) with the energy operator H = −∆Ω (in this Section we use the subscript
Ω in order to emphasize the dependence on the domain). The spectral analysis
of the Laplace operator
−∆Ω
in L2 (Ω)
with Dirichlet boundary conditions at ∂Ω has therefore attracted a lot of interest
in the last decades, [12, 14, 17, 21, 22, 31, 34, 39, 44, 68]. Considerable attention
was paid to the link between the geometry of Ω and the spectrum of −∆Ω . For
the unperturbed waveguide Ω0 it is easily seen that
σd (−∆Ω0 ) = ∅,
σess (−∆Ω0 ) = [λ1 , ∞),
(1.7)
where λ1 > 0 is the lowest eigenvalue of the Laplace operator on the cross-section
ω with the Dirichlet boundary conditions. If Ω results from a sufficiently local
perturbation of Ω0 , then the essential spectrum of −∆Ω coincides with that of
−∆Ω0 . In other words
σess (−∆Ω ) = [λ1 , ∞),
(1.8)
The discrete spectrum of −∆Ω0 , however, turns out to be unstable against
certain geometrical perturbations of Ω0 . In particular, it was proved in [34, 39,
21, 14] that an appropriate bending or a local enlargement of Ω0 lead to the
existence of a discrete spectrum of −∆Ω . Thus, the Laplace operator in the
bent or locally enlarged domain Ω has at least one discrete eigenvalue below λ1 .
The same situation of course occurs also in the straight waveguide Ω0 if we add
a negative potential perturbation to the Laplace operator −∆Ω0 .
The common feature of these effects is that the discrete eigenvalues appear
immediately, no matter how small the geometrical or potential perturbation
is. Hence, by the evolution laws of quantum mechanics, see Section 1.1, a local
bending or enlarging of the straight waveguide Ω0 leads to the existence of timeindependent states and consequently to a localisation of particles (electrons).
This phenomenon has of course a negative influence on the transport of energy
in Ω.
In [11, 27, 28, 47] we have shown how these undesired localisation effects
might be, up to a certain extent, suppressed by adding an appropriate magnetic
field or by an appropriate ”twisting” of the waveguide (see section 1.2 for the
definition of twisting). More precisely, we have proved that, in the presence of
a local magnetic field or in locally twisted waveguides, the discrete eigenvalues
of the corresponding Laplace operator do not appear after any attractive perturbation of −∆Ω0 , but only if this perturbation (of geometrical or potential
type) is strong enough. The key ingredient of the proof are certain Hardy-type
inequalities for the unperturbed Laplace operator. In the next section we will
describe the connection between such inequalities and the spectral properties of
Laplace operators at threshold of the essential spectrum.
Hardy inequalities and virtual bound states
Laplace operator in Rn . Consider a test function u ∈ C0∞ (Rn ) and suppose
18
that n ≥ 3. A standard argument using an integration by parts and the CauchySchwarz inequality shows that
Z
Z
(n − 2)2
|u|2
|∇u|2 dx
n ≥ 3.
(1.9)
dx
≤
4
|x|2
Rn
Rn
This inequality extends by density to all functions u ∈ H 1 (Rn ). Since H 1 (Rn )
is the form domain of the Laplace operator −∆ in L2 (Rn ), it follows from (1.9)
that
(n − 2)2
≥ 0
(1.10)
−∆ −
4|x|2
holds true in the sense of quadratic forms in L2 (Rn ) provided n ≥ 3. Now
assume that the Laplace operator is perturbed by a potential V which is sufficiently regular and decays at infinity. The essential spectrum of the resulting
Schrödinger operator
Hα = −∆ + α V
in
L2 (Rn ),
α ∈ R,
(1.11)
thus coincides with the half-line [0, ∞). If the potential V decays at least as
fast as |x|−2 at infinity, then inequality (1.9) implies that the discrete spectrum
of Hα remains empty for α small enough. Indeed, let us say for simplicity that
V is bounded and that V (x) = o(|x|−2 ), |x| → ∞. In view of (1.9), (1.10) we
obtain
(n − 2)2
(n − 2)2
+
+ αV ≥ 0
Hα = −∆ −
4|x|2
4|x|2
for α sufficiently small. It follows that for these values of α the operator Hα
has no negative spectrum and therefore no bound states.
Estimate (1.9) is a particular case of the so-called Hardy inequality for functions in H 1 (Rn ), which in general can be written as
Z
Z
|∇u|2 dx, ρ ≥ 0,
(1.12)
ρ |u|2 dx ≤
Rn
Rn
where ρ is called a Hardy weight, see [65]. As mentioned above, such an inequality implies certain stability of the spectrum of Schrödinger operators in L2 (Rn )
with n ≥ 3, in the sense that, for V vanishing at infinity and α small enough,
we have
σ(−∆) = σ(−∆ + αV ), n ≥ 3.
The situation in the case n ∈ {1, 2} is completely different. By a proper
choice of a sequence of test functions it can be shown, see [8], that, if n ≤ 2,
then (1.12) implies ρ ≡ 0. In other words, there exists no non-trivial weight ρ
for which the Hardy inequality holds true. In the language of spectral theory
of Schrödinger operators,
this reflects the well-known weak coupling behaviour
R
of Hα . Namely, if Rn V < 0, then the operator Hα has at least one negative
bound state for any α > 0, not only for α large enough, see [71].
19
Operators which produce at least one bound state below the threshold of
their essential spectrum after adding an arbitrarily small negative perturbation,
are in general called operators with a virtual bound state, see [75]. In the
sequel, we will often use this terminology also for other types of Schrödinger
operators. The expression “virtual bound state” is sometimes replaced with
“virtual level” or “resonance” at threshold, see [7, 24].
The connection between the validity of a Hardy inequality (1.12) and the
spectral properties of the operator Hα at threshold is now clear: either (1.12)
implies ρ ≡ 0, in which case −∆ has a virtual bound state at zero, or (1.12) holds
for some non-trivial ρ, which means that the virtual bound state is absent. We
will shortly discuss this issue in the case of Schrödinger operators in waveguides.
Before doing so, let us make several comments concerning inequality (1.9) and
its modifications.
Remark 1.
(i) The quadratic decay of the weight in (1.9) is independent of the dimension and cannot be improved. Indeed, the decay of the Hardy weight is
determined by the fact that −∆ is a second order differential operator,
see e.g. [8]. Moreover, the constant in (1.9) is sharp.
(ii) As mentioned above, the Hardy inequality fails in dimension one. However, if we consider a class of functions u ∈ H 1 (R) such that u(x0 ) = 0 for
some x0 ∈ R, then a simple integration by parts shows that
Z u′ −
R
u
2(x − x0 )
2
dx +
Z
R
|u|2
dx =
4(x − x0 )2
Z
|u′ |2 dx ,
R
which implies
Z
R
|u|2
dx ≤ 4
(x − x0 )2
Z
|u′ |2 dx,
u(x0 ) = 0.
(1.13)
R
This means that adding a Dirichlet boundary condition at a point x0 leads
to a Hardy-type inequality and hence removes the virtual bound state of
the Laplace operator in dimension one.
(iii) Numerous modifications and generalisations of (1.9) may be found in [65,
61].
Laplace operator in waveguides.R We consider the Laplace operator −∆Ω0
associated with the quadratic form Ω0 |∇ u|2 , defined on its domain H01 (Ω0 ).
From (1.7) we conclude that
Z
Z
2
|∇ u|2 dx
∀ u ∈ H01 (Ω0 ) .
λ1
|u| dx ≤
Ω0
Ω0
20
Inequality (1.12) thus becomes trivial in this situation since λ1 is strictly positive. This makes us address the question whether there exists a non-trivial
weight ρ ≥ 0 such that the modified Hardy inequality
Z
Z
2
ρ |u| dx ≤
|∇u|2 − λ1 |u|2 dx
(1.14)
Ω0
Ω0
holds true for all u ∈ H01 (R). Inequality (1.14) would lead to a stability of the
discrete spectrum of −∆Ω0 in the same way as in the case of the Laplace operator
in Rn . However, it is easy to see that (1.14) fails. Indeed, let x = (x1 , x′ ), where
x′ denotes the coordinate on the cross-section ω and let χ1 (x′ ) be the normalised
eigenfunction of −∆ in L2 (ω) associated with λ1 . Consider the sequence of test
functions
!
|x1 − |xx11 | n|
′
′
|x1 | > n.
un (x) = χ1 (x ) |x1 | ≤ n, un (x) = χ1 (x ) exp −
n
Inserting un into (1.14) we find out that
Zn Z
−n ω
ρ(x) |χ1 (x′ )|2 dx′ dx1 → 0
as
n → ∞.
Since χ1 can be chosen positive in ω, it follows that ρ = 0 a.e. This is of course
in agreement with the known facts (see e.g. [21, 34]) saying that the discrete
spectrum of −∆Ω0 is unstable in the sense that any attractive perturbation
induces the existence of discrete eigenvalues below λ1 . Put differently, −∆Ω0
has a virtual bound state at λ1 .
However, it turns out that this virtual bound state can be removed in certain
situations. Namely, in dimension two (i.e. ω ⊂ R), if
(i)
a local magnetic field is added to the system,
and in dimension three (i.e. ω ⊂ R2 ), if
(ii)
the waveguide Ω0 is appropriately twisted.
The latter holds provided ω is not rotationally symmetric. In other words, the
new operator, which is either the Laplacian with a magnetic field acting in Ω0
in case (i), or the usual Laplacian acting in the twisted waveguide in case (ii),
satisfies an inequality of the type (1.14). This means that in the presence of a
local magnetic field or in twisted waveguides, the bound states appear only if the
respective attractive perturbation is strong enough; roughly speaking, stronger
than the Hardy-weight ρ.
In the next two sections we will describe, after some preliminaries, how the
resulting weight function ρ in (1.14) depends on the magnetic field, respectively
on the twisting of the waveguide.
21
Waveguides with magnetic field
In most parts of Chapter 2 we will deal with two-dimensional waveguides, which
means Ω0 = R×(0, d) with d > 0. Correspondingly, the threshold of the essential
spectrum of −∆Ω0 is equal to
π2
λ1 = 2 .
d
As mentioned above, we want to focus on the case in which a local magnetic
field is present in the waveguide. It will be therefore useful to say a few words
on magnetic Schrödinger operators in two dimensions.
Schrödinger operators with magnetic field. Let A = (a1 (x), a2 (x)) be a
real valued vector field in R2 such that both a1 and a2 are in L2loc (R2 ). Introduce
the quadratic form
Z
|(−i∇ + A) u|2 dx
(1.15)
R2
on C0∞ (R2 ). The closure of this form induces the operator
HA = (−i∇ + A)2
acting in L2 (R2 ). If a function B : R2 → R satisfies
B(x) =
∂a1
∂a2
(x) −
(x)
∂x1
∂x2
∀ x ∈ R2 ,
then HA is the quantum mechanical energy operator of a charged particle moving in a two-dimensional plane and subject to a perpendicular magnetic field
of intensity B(x). If B decays at infinity in some average sense, then it can be
shown that
σess (HA ) = [0, ∞) ,
which means that adding a (local) magnetic field B does not affect the essential
spectrum of H0 = −∆.
On the other hand, it was noted by Laptev and Weidl, see [54], that the
presence of a magnetic field removes the virtual bound state of −∆ at zero.
Namely, it was proved in [54] that, while any inequality of the form (1.12)
fails in R2 , it holds true if we replace the gradient ∇ by the magnetic gradient
(−i∇ + A). More precisely, given a vector potential A generating a non trivial
magnetic field B, sufficiently regular and decaying at infinity, there exists a
constant C(A) such that
Z
Z
|u|2
|(−i∇ + A)u|2 dx
(1.16)
dx ≤ C(A)
1 + |x|2
R2
R2
holds for all u ∈ C0∞ (R2 ). In the case of the Aharonov-Bohm magnetic field
which is generated by the potential
−x2 x1
, β ∈ R \ Z,
,
Aβ (x) = β
|x|2 |x|2
22
the Hardy inequality (1.16) takes the form
Z
Z
|u|2
|(−i∇ + Aβ )u|2 dx ∀ u ∈ C0∞ (R2 \ {0}) .
dx ≤ C(β)
|x|2
R2
(1.17)
R2
−2
Moreover, the constant C(β) = (mink∈Z |k − β|) is sharp in this case. Recall
that β is the flux of the Aharonov-Bohm field. Further generalisations of Hardytype inequalities for magnetic Dirichlet forms have been established in [1, 3].
Magnetic Schrödinger operators in waveguides (Section 2.1). In [27]
we consider the operator
Ω0
HA
= (−i∇ + A)2
associated with the closure of the form
Z
|(−i∇ + A) u|2 dx,
in L2 (Ω0 ),
u ∈ C0∞ (Ω).
(1.18)
Ω0
Our main result, see Section 2.1, extends inequality (1.16) to quantum waveguides. In other words, it shows that inequality (1.14) holds true provided we
replace the gradient by (−i∇ + A). More precisely, under some localisation
conditions on magnetic field we show that
Z
Z |u|2
π2 2
2
dx
(1.19)
|(−i∇ + A)u| − 2 |u|
dx ≤ CA
1 + |x1 |2
d
Ω0
Ω0
for some constant CA > 0 and for all u ∈ C0∞ (Ω0 ). In the special case of the
Aharonov-Bohm field with the flux β the inequality takes the form
Z Z
π2 2
|u|2
2
dx ∀ u ∈ C0∞ (Ω0 \ {0}) .
dx ≤ Cβ
|(−i∇ + Aβ )u| − 2 |u|
|x|2
d
Ω0
Ω0
(1.20)
Inequalities (1.19) and (1.20) show that adding a local magnetic field (or the
Aharonov-Bohm field) to the waveguide removes the virtual bound state of the
2
associated Schrödinger operator at the threshold πd2 .
As an application of these inequalities, we obtain the stability of the specΩ0
trum of HA
against weak attractive perturbations. In particular, we prove in
Section 2.1 that a weak bending or a weak local enlargement of Ω0 will not
induce any bound states in the presence of the magnetic field, contrary to the
non-magnetic case.
Another type of attractive perturbation which induces bound states of the
non-magnetic Laplacian is a local change of the boundary condition from Dirichlet to Neumann, which we discuss next.
Perturbation of boundary conditions (Section 2.2). Apart from the
geometrical perturbations of Ω0 , it has been shown, see [14, 35, 36, 38, 66], that
23
bound states appear also in waveguides with the so-called “Neumann window”.
The Neumann window is represented by a compact segment of the boundary of
Ω0 , on which Dirichlet boundary conditions are replaced by Neumann boundary
conditions. This is again a type of attractive perturbation which does not affect
the essential spectrum, but produces bound states no matter small the lenght
of the window is.
It is thus natural to address the question whether an appropriate magnetic
field might suppress this effect as well. The answer is positive, although one
cannot simply carry over the technique of [27] to this case, because changing
the boundary conditions is a very subtle perturbation. Namely, it is a stronger
perturbation than enlarging Ω0 (which means adding a “bump”), in the following sense: existence of a bound state in a waveguide with a bump added to a
segment of the boundary implies the existence of a bound state in a waveguide
with a Neumann conditions on that segment, see [14].
Moreover, introducing a magnetic field to a waveguide with a Neumann
window on a segment γ ⊂ ∂Ω0 actually means also changing the boundary
conditions on γ to the so-called magnetic Neumann conditions. The latter
require that
(1.21)
(−i∂x2 + a2 )u(x) = 0 on γ ,
where a2 is the second component of the vector potential A. Therefore, we
cannot use inequality (1.19), since it holds for a different class of test functions.
Consequently, one has to establish a modified version of (1.19), which takes into
account boundary condition (1.21).
We have done this in [11], and proved the result stating that the bound
states in the waveguide with an appropriate magnetic field will not appear if
the width of the segment γ is small enough, see Section 2.2 for details.
Twisted waveguides
The effect of a local magnetic field in two-dimensional waveguides, i.e. removing the virtual bound state at the threshold of the essential spectrum, can be
achieved in the absence of a magnetic field as well, by a so-called twisting of the
waveguide Ω0 . We distinguish two types of twisting.
Geometrical twisting (Section 2.3). In order to define what a twisting
means, let us consider a three- dimensional waveguide Ω0 = R × ω, where the
cross-section ω is an open bounded and connected subset of R2 . Given a smooth
function θ on R, we define the matrix valued function


1
0
0
Θ(x1 ) = 0 cos θ(x1 ) − sin θ(x1 ) ∀ x1 ∈ R
(1.22)
0 sin θ(x1 ) cos θ(x1 )
and the transformed waveguide Ω ⊂ R3 by
Ω = {Θ(x1 ) x : x = (x1 , x2 , x3 ) ∈ Ω0 } .
24
(1.23)
z
z
x
x
y
y
Figure 1.1: On the left, the plot of the surface of a rectangular waveguide
without twisting; in the right, the plot of the surface of the twisted rectangular
waveguide. The bold line represents the boundary of ω.
The new waveguide Ω clearly results from Ω0 by rotating the cross-section ω by
the angle θ(x1 ), which depends on the longitudinal coordinate x1 . We say that
dθ
is not identically zero. An example of a
Ω is twisted if the derivative θ̇ := dx
1
twisted waveguide with a rectangular cross-section ω is plotted in Figure 1.1.
If θ̇ has a compact support, then
σ (−∆Ω ) = σess (−∆Ω ) = σ (−∆Ω0 ) = [λ1 , ∞) ,
(1.24)
where, as usual, λ1 denotes the lowest eigenvalue of −∆ in L2 (ω) with Dirichlet
boundary conditions. Hence, a local twisting does not change the spectrum of
−∆Ω0 .
It does, however, remove the virtual bound state of −∆Ω0 at the threshold
λ1 . This is proved in [28], see Section 2.3. In fact, we have shown that inequality
(1.14) holds true with some non-zero ρ ≥ 0, provided we replace Ω0 by the
twisted waveguide Ω. The rusulting weight function ρ of course depends on the
geometry of ω and on the twisting function θ̇. The former is reflected through
the number
λ :=
inf1
k∇ϕk2L2 (ω) − λ1 kϕk2L2 (ω) + k(x2 ∂x3 − x3 ∂x2 ) ϕk2L2 (ω)
kϕkL2 (ω)
ϕ∈H0 (ω)
.
(1.25)
Note that, in view of (1.24), we have λ ≥ 0. For compactly supported θ̇ the
resulting Hardy-type inequality reads as follows, see Section 2.3,
Z
Z
c(θ̇)
λ |θ̇|2 |u|2 ≤
∀ u ∈ H01 (Ω) ,
(1.26)
|∇u|2 − λ1 |u|2
Ω
Ω
25
where c(θ̇) is a positive constant depending on θ̇. Since the last term in the
numerator of (1.25) vanishes if and only if ϕ is radially symmetric on ω it can
be shown (see [28]) that
λ=0
⇐⇒
ω is rotationally symmetric
Hence, roughly speaking the value of λ tells us how much the cross-section
ω differs from a rotationally symmetric domain. If ω is a disc, for example,
then the infimum in (1.25) is attained at the eigenfunction of the operator −∆
in L2 (ω) associated with λ1 , which gives λ = 0. As expected, in this case
inequality (1.26) becomes trivial. This also explains the existence of a virtual
bound state of the Laplace operator in twisted tubes with circular cross-sections
observed in [17, 21, 41].
On the other hand, as long as ω is non-symmetric, λ is strictly positive and a
local twisting will remove the virtual bound state of −∆Ω0 . In the same way as
in the case of two-dimensional waveguides with magnetic field, we have applied
inequality (1.26) to prove that an appropriate twisting removes also bound states
induced by sufficiently weak perturbations of −∆Ω , see Section 2.3. In this
sense, we might say that the transport of energy in twisted waveguides with
non circular cross-sections is more stable, compared to non twisted waveguides.
Remark 2.
(i) It is possible to derive a lower bound on λ in some particular situations,
such as ω being a rectangle or a square, but the general case remains an
open problem.
(ii) The constant c(θ̇) in (1.26) is for large θ̇ inversely proportional to kθ̇k2∞ .
This is not surprising, since we cannot increase the weight function on the
left-hand side of (1.26) arbitrarily by increasing θ̇, see [28].
(iii) The repulsive effect of twisting was observed, in a different setting, also in
[13, 40, 41]. The behaviour of embedded eigenvalues in twisted waveguides
has been recently inspected in [48].
Twisting of boundary conditions (Section 2.4). The reason why the
virtual bound state of −∆Ω0 disappears when Ω0 gets twisted is that twisting
breaks the translational invariance of Ω0 . A similar situation might occur also
in a two-dimensional waveguide with combined Dirichlet-Neumann boundary
conditions.
If Ω0 ⊂ R2 is a two-dimensional strip, then of course it cannot be twisted geometrically as in the three-dimensional case. Nevertheless, it might be “twisted”
through a switch of boundary conditions, provided these are different at the two
opposite parts of the boundary of Ω0 . This is shown in Section 2.4.
Let Ω0 be the strip of width d and let −∆0 be the Laplace operator in Ω0
with a Neumann boundary condition at the upper boundary and a Dirichlet
26
Figure 1.2: The lower waveguide results by a switch of Dirichlet (thick lines) to
Neumann (thin lines) boundary conditions, and vice versa.
boundary condition at the lower boundary, see Figure 1.2 in the top. The
infimum of the spectrum of −∆0 can be easily calculated, and we arrive at
2
π
σ(−∆0 ) = σess (−∆0 ) =
,∞ .
4 d2
As in the case of the strip with purely Dirichlet boundary conditions, −∆0 has
2
a virtual bound state at the threshold 4πd2 . This means that there exists no
weight function ρ ≥ 0, ρ 6≡ 0 such that the inequality
Z Z
π2
2
2
2
,
(1.27)
|u|
|∇u| −
ρ |u| ≤
4 d2
Ω0
Ω0
holds true for all test functions u in the form domain D0 of −∆0 .
Nevertheless, if we switch the boundary conditions at a certain point from
Dirichlet to Neumann and vice versa, see Figure 1.2, then the functions in the
new form domain D will satisfy inequality (1.27) with some non-trivial ρ. In
other words, (1.27) holds true for the test functions which respect the change
of the boundary conditions. We have proved the latter in [47], see Section 2.4.
As a consequence, we find out that, similarly as the geometrical twisting in
dimension three, the “twisting” of the boundary conditions prevents the existence of bound states in the presence of sufficiently weak negative perturbations.
In other words, while in the waveguide showed in the top of Figure 1.2 any negative perturbation of the Laplace operator induces a discrete eigenvalue below
π2
4 d2 , in the “twisted” waveguide (Figure 1.2 in the bottom) this will happen
only if such a perturbation is strong enough.
A particular case of a negative perturbation is the so-called opening of the
“Neumann window”: the switching point (from Neumann to Dirichlet) in the
upper part of the boundary, see Figure 1.2, is moved to the right by some ε > 0.
We thus obtain a “Neumann window” of a width ε. In this case, the absence
of bound states for small values of ε was proved already in [20] by a direct, but
tedious, estimation of the corresponding quadratic form. Inequality (1.27) thus
shows that there is a deeper reason behind the effect observed in [20], and that
27
the result holds also for other types of (local) perturbations. Namely, the switch
of the boundary conditions destroys the translational invariance and removes
the virtual bound state at the threshold π 2 /4d2 .
Periodically twisted waveguide (Section 2.5). Before closing this section,
let us briefly go back to the three-dimensional case and discuss a model of a
periodically twisted waveguide which we have studied in [32]. Periodical twisting
means that the twisting function θ̇ is constant, say β = θ̇ . In this situation,
contrary to a local twisting the threshold of the essential spectrum of the Laplace
operator in a twisted tube Ω is not the same as in the non twisted tube Ω0 , but
is strictly larger, i.e.
κ := inf σess (−∆Ω ) > inf σess (−∆Ω0 ) .
The actual value of κ depends on β, see Section 2.5 for details. It turns out that
a local perturbation of this periodical twisting will induce the existence of bound
states below the new threshold of the essential spectrum. Indeed, if we replace
the constant twisting β by θ̇(x1 ) = β − ε(x1 ), where ε is a compactly supported
function, then the result in [32] states that the bound states will appear if
Z
(θ̇2 (x1 ) − β 2 ) dx1 < 0 .
R
This condition is similar to the one needed for the existence of bound states in
locally enlarged waveguides, see [14, 36].
1.3
Spectral estimates
In Chapter 3, we will deal with the spectral properties of Laplace and Schrödinger
operators on a special class of graphs, called metric trees. Before we start with
the discussion of metric trees, it will be convenient to give a brief description of
the known results on negative eigenvalues of Schrödinger operators in Euclidean
spaces.
Discrete spectrum in Rn
Let V a real valued potential, α a positive coupling constant and let Hα be the
Schrödinger operator
−∆ − αV
(1.28)
in L2 (Rn ). For simplicity, assume that V ≥ 0. If V decays at infinity, then the
general results concerning the stability of the essential spectrum of self-adjoint
operators, see [8, 67] say that
σess (−∆ − αV ) = σess (−∆) = [0, ∞) ∀ α ∈ R .
Hence, the negative spectrum of −∆ − αV consists of discrete eigenvalues Ej
of finite multiplicity. The task of spectral theory is to find a link between
28
these eigenvalues and the potential term αV in (1.28). Since it is in general
impossible to find individual estimates on each Ej , one would like to obtain
some information, for example, on the so-called Riesz means:
X
|Ej |γ for γ > 0,
tr (−∆ − αV )γ− =
j
N (−∆ − αV ) =
X
1
for γ = 0.
(1.29)
j:Ej <0
Here, tr(T )− stands for the trace of a negative part of an operator T and N (T )
for the number of negative eigenvalues of T (counting multiplicities). It is of
course not a priori clear whether the sums in (1.29) are finite or infinite and, in
the latter case, whether they converge or not. This depends on the regularity of
V and its behaviour at infinity. It is illustrative to look first at the asymptotics
of (1.29) in the limit α → ∞.
Large coupling. In order to study the asymptitical behaviour of N (−∆−αV ),
it is convenient to apply the technique known as Dirichlet-Neumann bracketing,
which was developed in [18, Chap. 6], see also [67]. For continuous potentials
with compact support, an appropriate application of this method (see e.g. [67,
Chap. XIII.15]) gives
n Z
n
ωn α 2
n
α → ∞,
(1.30)
N (−∆ − αV ) =
V 2 (x) dx + o α 2
n
(2π)
Rn
where ωn denotes the volume of the unit ball in Rn . Note that the right-hand
side of (1.30) equals, up to the factor (2π)n , the volume of the classical phase
space Rn × Rn , where the classical Hamiltonian symbol q(x, ξ) = |ξ|2 − αV (x)
is negative:
Z
Z
n
n
V 2 (x) dx .
dx dξ = ωn α 2
Rn
(x,ξ)∈R2n :q(x,ξ)<0
In the same way, using the bracketing technique, one can derive the asymptotics
of (1.29) also for γ > 0, which leads to
Z
n
n
γ
cl
γ+ n
2
V γ+ 2 (x) dx + o αγ+ 2
α → ∞, (1.31)
tr (−∆ − αV )− = Lγ,n α
Rn
with
Lcl
γ,n =
Γ(γ + 1)
.
2n π n/2 Γ γ + 1 + n2
(1.32)
A sufficient condition for (1.31) to hold is again that V is continuous and compactly supported. The question is whether (1.30) might be extended to all
potentials for which the integral on the right-hand side converges. In dimensions n ≤ 2, the answer is “no”. For n = 1, this was shown in [64]. The
two-dimensional case is discussed in Section 1.4. In dimensions n ≥ 3, on the
29
contrary, one can indeed extend (1.30) to all potentials V ∈ Ln/2 (Rn ). To this
end, one needs a suitable uniform upper bound on N (−∆ − αV ).
Bounds on the number of negative eigenvalues. The problem how to
estimate the number of negative eigenvalues of −∆−αV in terms of the potential
V was addressed already by Bargmann in [4]. Although he studied the threedimensional case with spherically symmetric potentials, his results can be easily
carried over to dimension one, i.e.
Z
d2
N − 2 − αV ≤ 1 + α
|x| V (x) dx .
(1.33)
dx
R
The constant term 1 cannot be removed, due to the existence of a virtual bound
state at zero.
A general and very useful technique for studying N (−∆−αV ) was developed
in [6, 70] by Birman and Schwinger. The result of [6] and [70] says that for any
t ≥ 0 the number of eigenvalues of −∆ − αV below −t ≤ 0 equals the number
of eigenvalues of the operator
αV
1
2
(−∆ + t)−1 V
1
2
(1.34)
above 1. This remarkable fact is the well-known Birman-Schwinger principle. If
V decays at infinity, then (1.34) is compact and the number of its eigenvalues
which are larger than 1 can be estimated, for example, by the square of its
Hilbert-Schmidt norm, see [8, 10]. In dimension n = 3, this leads to the BirmanSchwinger bound
Z Z
V (x) V (y)
α2
dxdy .
(1.35)
N (−∆ − αV ) ≤
2
16 π
|x − y|2
R3 R3
However, both the Birman-Schwinger bound and the Bargmann bound (1.33)
have a common artefact: in the large coupling limit α → ∞, they grow faster
in α then the asymptotics in (1.30). It is thus natural to ask whether one
can estimate N (−∆ − αV ) by an upper bound which reflects its asymptotical
behaviour for large α. This was answered independently by Cwikel, Lieb and
Rozenbljum, who proved that
Z
n
n
N (−∆ − αV ) ≤ C(n) α 2
V 2 (x) dx for n ≥ 3
(1.36)
Rn
see [19, 58, 69]. The constant C(n) in (1.36) depends on the dimension, but
n
is uniform with respect to all potentials V ∈ L 2 (Rn ). Estimate (1.36) is one
of the most famous results of the spectral theory of Schrödinger operators and
is commonly known as the Cwikel-Lieb-Rozenbljum inequality. It also allows
n
one to extend the asymptotical law (1.30) to all V ∈ L 2 (Rn ), see [69]. Another
proof of (1.36) was found in [57]. An interesting generalisation of [57] for Markov
generators was obtained in [56].
30
Note that (1.36) fails to hold in dimensions n = 1 and n = 2. Indeed, due
to the appearance of a virtual bound state we have N (−∆ − αV ) ≥ 1 for any
α > 0, which clearly contradicts (1.36). We will see that suitable estimates on
(1.29) can be extended also to lower dimensions, provided the power γ is large
enough.
Lieb-Thirring inequalities. In 1976 Lieb and Thirring proved that the Riesz
means (1.29) can be estimated as follows
Z
n
n
γ
tr (−∆ − αV )− ≤ Lγ,n αγ+ 2
V γ+ 2 (x) dx ,
(1.37)
Rn
with some constant Lγ,n independent of V , see [59]. This inequality holds true
provided
γ ≥ 1/2 if n = 1,
γ > 0 if
n = 2,
γ≥0
if n ≥ 3.
(1.38)
Apart from being an interesting theoretical result on its own, inequality (1.37)
has been successfully applied in the proofs of the stability of matter in different
quantum systems.
The original method of [59], however, does not work in the critical cases
γ = 0 in dimension n ≥ 3, and γ = 1/2 in dimension n = 1. The former is
covered by the Cwikel-Lieb-Rozenbljum bound discussed above, while the latter
was solved by Weidl in 1996, see [76].
As for the constant on the right-hand side of (1.37), we point out that (1.31)
implies Lγ,n ≥ Lcl
γ,n . Sharp values of Lγ,n are only known in certain cases. In
[43] it was proved that
1
,
L1/2,1 = 2 Lcl
1/2,1 =
2
see also [42]. For higher values of γ we have
Lγ,n = Lcl
γ,n
∀γ ≥
3
,
2
∀n ∈ N,
which was proved by Laptev and Weidl in [54]. In this context, we recall the
conjecture
Lγ,n = Lcl
∀ γ ≥ 1, ∀ n ≥ 3
γ,n
stated by Lieb and Thirring in [59].
Let us finally mention an interesting observation which concerns inequality
(1.37) in dimensions n ≥ 3. Namely, in view of (1.9) we find that
γ
(n − 2)2
tr −∆ −
=0
∀ γ ≥ 0 ∀ n ≥ 3.
4|x|2
−
On the other hand, for V = |x|−2 the r.h.s. of (1.37) equals infinity for any n
and γ. This means that, in this situation, inequality (1.37) gives an estimate
31
which is certainly far from optimal. This problem has been recently pointed out
by Ekholm and Frank, who improved (1.37) by showing that
γ
tr (−∆ − αV )− ≤ Cγ,n
γ+ n2
Z (n − 2)2
dx ∀ γ > 0, ∀ n ≥ 3,
αV (x) −
4|x|2
+
Rn
see [24], and [25] for the one-dimensional case.
Metric trees
We understand a metric tree as the union of a set of (countably many) vertices and a set of edges, which are one-dimensional intervals connecting the
vertices (see below for details). Metric trees form a special class of quantum
graphs, which serve as mathematical models for various nano-technological devices. Different aspects of the spectral theory of Schrödinger and Laplace operators on metric graphs and trees have recently been studied in several works,
see e.g. [15, 30, 51, 48, 63, 62, 72, 73].
We want to focus on the connection between the spectral properties of
Schrödinger operators and the global geometry of a given tree. More precisely,
our aim is to establish suitable spectral estimates on metric trees analogous to
those in the Euclidean spaces described above. First, we need some prerequisites.
Laplace operator on a metric tree. A rooted metric tree Γ consists of a
set of edges E(Γ) and a set of vertices V(Γ). Given a point x ∈ Γ, we denote
by |x| the distance between x and the root o. If z ∈ V(Γ) is a vertex, then
its branching number b(z) is defined as the number of edges emanating from z.
We assume the natural conditions that b(z) > 1 for any vertex z 6= o and that
b(o) = 1. For a vertex z ∈ V(Γ) we define its generation gen(z) as the number
of vertices (including o) which lie on the unique path connecting z with the
root o. We will confine ourselves to the analysis of the so called regular trees
introduced in [62]. Regular trees are trees for which all the vertices of the same
generation have the same branching number and all the edges emanating from
these vertices have the same length.
We define the Neumann Laplacian −∆N as the self-adjoint operator in L2 (Γ)
associated with the closed quadratic form
Z
|ϕ′ (x)|2 dx, ϕ ∈ H 1 (Γ).
Γ
The notation is justified by the fact that the functions in the domain of −∆N
satisfy the Neumann boundary condition at o, see [26, 73]. We assume that
the potential V (x) on Γ depends only on |x| and, for simplicity, that V is nonnegative. We are interested in Schrödinger operator
in L2 (Γ).
−∆N − V
32
In the previous section we have seen that the character of the spectral estimates
for Schrödinger operators in Rn highly depends on the spatial dimension n. The
latter can also be expressed through the dependence of the surface of a given
ball on its radius. In order to carry over this concept to metric trees, we need
the notion of the so-called global dimension of Γ, which was introduced in [46].
To this end, we consider the branching function g0 : R+ → N defined by
g0 (t) = #{x ∈ Γ : |x| = t} .
(1.39)
The significance of the branching function is obvious: g0 tells us how fast the
surface of the ball {x ∈ Γ : |x| ≤ t} grows as a function of its radius t. Therefore,
we will say that Γ has global dimension d ≥ 1 if there exist positive constants
c1 , c2 such that
c1 ≤
g0 (t)
≤ c2
(1 + t)d−1
∀ t ∈ R+ .
(1.40)
On the other hand, any metric tree is locally one-dimensional. We thus have
a structure with a mixed type of dimensionality and, as we will see in Sections
3.1 and 3.2, the spectral properties of −∆N − V depend on the ratio between
the local and the global dimension of Γ.
Remark 3. Note that d need not be an integer, and that the class of metric
trees which possess a global dimension is quite rich. Nevertheless, there certainly
exist metric trees with no global dimension, which means that (1.40) represents
an additional assumption on the geometry of Γ.
Let us for the sake of brevity assume from now on that supe∈E(Γ) |e| = ∞ (in
Section [26] we also consider the case when the latter is finite). Under this
condition, it was proved in [73] that the spectrum of −∆N is purely essential
and covers the positive half-line. Consequently, for V that vanishes at infinity
we have
σess (−∆N − αV ) = [0, ∞) ∀ α ≥ 0
and the negative spectrum of −∆N − αV consists of discrete eigenvalues only.
Weak coupling behaviour (Section 3.1)
The first question that one has to answer when studying the behaviour of −∆N −
αV in the weak coupling limit α → 0, is whether there is a virtual bound state
at zero or not. It turns out that this is completely determined by the behaviour
of g0 at infinity, see [26, 63]. Namely, if the reduced height
LΓ :=
Z∞
0
dt
g0 (t)
(1.41)
of Γ is finite, then there is no virtual bound state and the negative spectrum of
−∆N − αV remains empty for α small enough.
33
If, on the contrary, g0 grows too slowly so that the integral in (1.41) diverges,
then −∆N −αV has at least one eigenvalue for any α > 0. In order to determine
the asymptotics of the lowest eigenvalue E1 (α) as α → 0, we assume that Γ has
a global dimension d. For LΓ = ∞ to hold it is necessary that d ∈ [1, 2].
We have proved in [46], see Section 3.1, that for α small enough this eigenvalue is unique and satisfies
2
|E1 (α)| ∼ α 2−d
for 1 ≤ d < 2 ,
−1
|E1 (α)| ∼ e−cα
for
(1.42)
d = 2,
where c > 0. It is interesting to compare this result with the known asymptotical
formulae in Euclidean spaces established in [71]:
n=1:
|E1 (α)| ∼ α2 ,
n=2:
−1
|E1 (α)| ∼ e−cα .
(1.43)
If the global dimension d equals 1, then (1.42) agrees, in the order of α, with
n = 1 in (1.43). In the particular case of the so-called branching graphs, i.e. one
vertex and finitely many edges, the precise asymptotics for E1 (α) was found
in [30].
As d grows, E1 (α) goes faster to zero, and, when d approaches the critical
value 2, the dependence becomes exponential as in (1.43). It follows that the
behaviour of E1 (α) for small α has nothing to do the with the local dimension
of Γ, but is completely determined by its global dimension, in other words by
the growth of g0 at infinity.
Weighted Lieb-Thirring inequalities (Section 3.2)
In [26] we have established eigenvalue estimates for −∆N − V on Γ analogous
to the Lieb-Thirring inequality in Rn . We distinguish between two main cases
according to whether the reduced height (1.41) is finite or infinite.
Case LΓ < ∞. Bounds on the number of eigenvalues. It is shown in
[26] that in this case the number of negative eigenvalues of −∆N − V can be
estimated by a weighted integral of V p , for some p ≥ 1. The power p and
the weight in the integral are linked through the branching function g0 in the
following way. If w : R+ → R+ is a positive function such that
 t
2/q ∞
Z
Z
q/2
−(q−2)/2


M = sup
g0 (s) w(s)
ds
g0 (s)−1 ds < ∞
(1.44)
t>0
t
0
for some q ∈ [1, ∞], then
N (−∆N − V ) ≤ C(Γ, q)
Z
V (x)p w(|x|) dx
(1.45)
Γ
q
q−2
p
and C(Γ, q) ∼ M , see Chapter 3 for details. In case the
with p =
q = ∞, the first integral on the left-hand side of (1.44) is understood as
34
sup0<s<t g0 (s)/w(s), and the constant C(Γ, ∞) = M is then sharp. We emphasise that (1.45) also holds for trees which do not have a global dimension. For
trees with a global dimension d is the condition LΓ < ∞ equivalent to d > 2.
Similar estimates with weighted integrals of V have been proved in [62]
for the eigenvalues of the operator (−∆)−1 V on Γ, with a Dirichlet boundary
condition at the root o.
Case LΓ = ∞. Weighted Lieb-Thirring estimates. If the reduced height
LΓ becomes infinite, then (1.45) fails to hold, or, in other words, C(Γ, q) = ∞.
This is due to the presence of the virtual bound state at zero. The corresponding
Riesz means tr(−∆N −V )γ− can be thus estimated in terms of V only for γ larger
than or equal to some positive minimal value. In order to find this value, we
need to assume that Γ has a global dimension d. Since LΓ = ∞, we have
d ∈ [1, 2] and the result of [26] reads as follows: for any a ≥ 0 and γ ≥ 0 the
inequality
Z
a+1
(1.46)
tr(−∆N − V )γ− ≤ C(γ, a, Γ) V (x)γ+ 2 (1 + |x|)a dx
Γ
holds true for
1−a
2
(1 + a)(2 − d)
γ>
2d
1−a
γ≥
2
γ>0
γ≥
if a ≤ d − 1 and 1 ≤ d < 2,
if a > d − 1 and 1 ≤ d < 2,
if
a < 1 and d = 2,
if
a ≥ 1 and d = 2,
Note that for 1 ≤ d < 2 and a = d − 1 inequality (1.46) also holds for the
minimal possible value of γ which is equal to 2−d
2 . In that case, (1.46) takes the
form
Z
2−d
V (x)(1 + |x|)d−1 dx,
tr(−∆N − V )−2 ≤ C
Γ
which for small potentials V reflects the weak coupling asymptotics given by (1.42).
For trees with d = 1 this inequality delivers an upper bound for the sum of the
square roots of eigenvalues of −∆N − V in the same way as in the Euclidean
case, see [76].
Weak versus strong coupling. It has been already mentioned that the spectral properties of −∆N − αV for α → 0 are determined by the global dimension of Γ. It is interesting to make a comparison with the large coupling limit
α → ∞. Let us assume that V is continuous and with compact support. Then,
the Dirichlet-Neumann bracketing technique gives
Z
γ+ 21
γ+ 21
γ+ 12
α
tr(−∆N − αV )γ− = Lcl
,
(1.47)
dx
+
o
α
V
(x)
γ,1
Γ
35
with the classical constant Lcl
γ,1 given by (1.32). It follows from (1.47) that
the large coupling behaviour of tr(−∆N − αV )− is one-dimensional and fully
independent of the global dimension d.
This discrepancy between the weak and the strong coupling can be heuristically understood by the following argument. If α is very small, then the
eigenfunctions associated with the eigenvalues of −∆N − αV have to be very
“flat” in order to make the quadratic form
Z
|ϕ′ (x)|2 − α V |ϕ|2 dx
Γ
negative. As α tends to zero the eigenfunctions are more and more spread
around the tree Γ, whose structure at infinity thus plays a crucial role. On the
other hand, for large values of α most of the eigenfunctions of −∆N − αV are
strongly concentrated around the support of V , and therefore do not “feel” the
global structure of Γ.
1.4
Two-dimensional Schrödinger operators
It is well-known that two-dimensional Schrödinger operators −∆ − α V exhibit
certain peculiar properties which bring about considerable difficulties in their
analysis. In Section 1.3 it was explained that the Cwikel-Lieb-Rozenbljum inequality (1.36) fails in R2 because of the presence of a virtual bound state.
However, this is not the only reason why (1.36) cannot hold for n = 2.
Namely, there are potentials that belong to L1 (R2 ), but for which the asymptotics of N (−∆− αV ) is non-regular as α → ∞, see [7] (by non-regular we mean
different from the one given in (1.30)). More exactly, one can construct functions V ∈ L1 (R2 ) so that N (−∆ − αV ) grows super-linearly in α. Of course,
this would be in contradiction with (1.36) for n = 2. These potentials can be
divided into two classes, according to the different origin of the non regular
behaviour of N (−∆ − αV ). The following canonical examples are taken from
[7].
Example 1: The threshold effect. Let σ > 1 and let V : R2 → R be a
spherically symmetric function defined by
Vσ (r) = r−2 (ln r)−2 (ln ln r)−1/σ
for r > e2 ,
Vσ (r) = 0
for r ≤ e2 .
(1.48)
It it easy to check that Vσ ∈ L1 (R2 ) for any σ > 1. However, it is shown in [7]
that
N (−∆ − αVσ ) ∼ ασ
as α → ∞ .
(1.49)
Choosing σ large enough, we can then achieve an arbitrary power-like growth
of N (−∆ − αVσ ). The origin of this effect is in the slow decay of Vσ at infinity.
As a consequence, the operator −∆ − αVσ has for large α too many negative
36
eigenvalues which are very close to the threshold zero (roughly speaking, the
eigenfunctions have enough room to spread over the portion of space where Vσ
is small but positive). Since all the negative eigenvalues contribute with 1 to
the sum N (−∆ − αVσ ), the presence of a large number of eigenvalues close to
zero leads to the super-linear growth in (1.49).
This effect disappears if, instead of counting all negative eigenvalues, we
count only those eigenvalues which lie below some constant −t < 0. The corresponding asymptotics for large α is then again regular:
N (−∆ − αVσ + t) ∼ α
as α → ∞ ∀ t > 0,
see [7].
Example 2: The effect of a local singularity. The super-linear asymptotics
(1.49) can also occur for potentials which are compactly supported, but have a
strong local singularity. An example would be
Vσ (r) = r−2 | ln r|−2 | ln | ln r||−1/σ
Vσ (r) = 0
for r < e−2 ,
for r ≥ e−2 .
(1.50)
Also in this case we have Vσ ∈ L1 (R2 ) for any σ > 1, but N (−∆ − αVσ ) is proportional to ασ when α → ∞, see [7]. The origin of the super-linear growth of
N (−∆ − αVσ ) is however completely different from the previous example. Here,
the potential well Vσ is arbitrarily deep in the vicinity of zero and can therefore
“accommodate” many mutually orthogonal eigenfunctions concentrated around
its support. As α tends to infinity, these eigenfunctions produce many eigenvalues which can be very far from zero. Consequently, we get
N (−∆ − αVσ + t) ∼ ασ
as
α→∞
∀t ≥ 0,
(1.51)
as was proved in [7]. The difference with respect to (1.49) is obvious: the
asymptotics is non-regular no matter where we pick the point −t.
It was shown in [52] that, if the potential V is spherically symmetric, then
these effects can be removed by adding a positive term b |x|−2 to the Laplacian.
More exactly, the inequality
Z
V (x) dx
(1.52)
N (−∆ + b |x|−2 − αV ) ≤ C(b) α
R2
holds true provided b > 0 and V (x) = V (|x|) ≥ 0. This is of course not in
contradiction with the above examples, since the functions Vσ in (1.48) and
(1.50) are dominated by the term b |x|−2 in the vicinity of infinity and zero,
respectively. A generalisation of (1.52) for non-symmetric potentials was proved
in [53]. The latter result was also applied in [2], where an upper bound on
the number of negative eigenvalues is proved for a two-dimensional Schrödinger
operator with an Aharonov-Bohm magnetic field. A very good survey of various
results and methods concerning estimates for the number of negative eigenvalues
of Schrödinger operators can be found in [9].
37
Logarithmic Lieb-Thirring inequalities (Section 4.1)
We have seen in Section 1.3 that, in dimension n = 1, the smallest γ for which
the Lieb-Thirring inequality (1.37) holds is equal to 1/2. This leads to
1/2 X q
Z
d2
=
tr − 2 − αV
|Ej | ≤ L1/2,1 α
V (x) dx ,
dx
−
j
(1.53)
R
2
d
where Ej are the negative eigenvalues of − dx
2 − αV , see [76, 43, 42]. In view
of (1.43) it is also clear that the power 1/2 in (1.53) cannot be replaced by a
smaller one, since this would lead to a contradiction for small α. Inequality
(1.53) thus can be understood as a certain borderline in dimension one.
In Section 4.1 we deal with the spectral estimates for −∆ − αV in dimension
two. Here, the situation is by far not as clear as in dimension one. First, the
infimum of all possible exponents γ in (1.37), which is zero, is not attained.
As already explained, for γ = 0 and n = 2 inequality (1.37) would lead to
inconsistencies for α → 0, because of the presence of a virtual bound state, as
well as for α → ∞, because of non-regular asymptotics of the type (1.49). On
the other hand, as soon as γ > 0, then the inequality
Z
X
|Ej |γ ≤ Lγ,2 αγ+1
V (x)γ+1 dx
(1.54)
tr (−∆ − αV )γ− =
j
R2
does not reflect the exponential asymptotic of the left-hand side in the limit
α → 0, see (1.43).
Motivated by this artefact of (1.54) we proved in [50] a new Lieb-Thirring
type inequality in two dimensions, in which we replaced the power function on
the l.h.s. of (1.54) by a family of functions Fs : (0, ∞) → (0, 1] defined by

0 < t ≤ e−1 s−2 ,
 | ln ts2 |−1
∀s > 0
Fs (t) :=
(1.55)

1
t > e−1 s−2 .
P
We note Fs (t) converges point-wise to 1 as s → ∞, which means that j Fs (|Ej |)
converges to the counting function N (−∆ − αV ). P
However, for each s > 0 we
have Fs (t) → 0 as t → 0. We thus might call
j Fs (|Ej |) a “regularised”
counting function. In order not to go too deep into technicalities we recall the
main result of [50] only for spherically symmetric potentials V , see Section 4.1
for the general statement.
If V (x) = V (|x|) for all x ∈ R2 , then there exist constants C1 and C2 such
that
X
Fs (|Ej |) ≤ C1 α kV ln |x/s|kL1 (B(s)) + C2 α kV kL1 (R2 )
(1.56)
j
holds for all s > 0, with B(s) = {x ∈ R2 : |x| < s}. The constants C1 and C2
are independent of s. Notice that the linear dependence of the r.h.s. of (1.56)
38
on α reflects the right asymptotics both for α → 0 and for α → ∞, see Section
4.1 for details.
It is also illustrative to confront (1.56) with the examples (1.48) and (1.50).
Namely, since the r.h.s. of (1.56) is finite for any Vσ of the type (1.48), it
follows that the super-linear
growth in (1.49) has been removed by replacing
P
N (−∆ − P
αVσ ) with j Fs (|Ej |). The reason for this is that the contribution to
the sum j Fs (|Ej |) of the eigenvalues close to the threshold is much smaller
than 1, and so threshold effect is suppressed.
On the other hand, it is clear that “regularising” the counting function
in the vicinity of zero by (1.55) cannot remove the non-regular asymptotics in
example (1.50), since the latter does not only result from eigenvalues close to the
threshold, see (1.51). In other words, example (1.50) shows that the unpleasant
term with the local logarithmic weight on the r.h.s. of (1.56) cannot be removed.
Estimates on the number of bound states (Section 4.1). The method
of [50] can also be easily applied to obtain estimates on N (−∆ − αV ). For
spherically symmetric potentials we then get
N (−∆ − αV ) ≤ 1 + α c0 kV ln |x/s|kL1 (R2 ) + kV kL1 (R2 )
∀ s > 0, (1.57)
for some c0 > 0. The general case is stated in Section 4.1. We emphasise that,
contrary to (1.56), here the function V must be integrable with the logarithmic
weight in the whole R2 , which excludes very slowly decaying potentials of the
type (1.48). A similar bound on N (−∆ − αV ) including a logarithmic weight
was proved in [74]. Note also, that since c0 does not depend on s, we can
actually optimise (1.57) by replacing the r.h.s. with its infimum over all s > 0.
This agrees with the bound established, with explicit constants, in [16].
Applications to quantum layers
Quantum layers might be regarded as the two-dimensional analogue of quantum
waveguides. We model a quantum layer of width d by a plate Ω = R2 × (0, d)
and the Laplace operator −∆Ω in L2 (Ω), with Dirichlet boundary conditions at
∂Ω. It is convenient to work with the shifted Hamilton operator
H0 = −∆Ω −
π2
d2
in
L2 (Ω).
(1.58)
The spectrum of H0 is then purely essential and covers the half-line [0, ∞).
Similarly as in the case of quantum waveguides, it turns out that bound states
(i.e. negative discrete eigenvalues) of H0 might be induced by an appropriate
geometrical deformation of Ω, such as bending or local enlarging (see [12, 22])
or simply by adding a negative potential V .
Inequality (1.56) and its general version established in Section 4.1 enables us
to find suitable spectral estimates on these bound states. We will describe the
results distinguishing the case of bound states induced by a potential perturbation of H0 and the case of bound states induced by a geometrical deformation
of the layer Ω.
39
Potential perturbations (Section 4.1). For x ∈ Ω we use the notation
x = (x1 , x2 , x3 ). Given a potential function V : Ω → R+ , let Ej be the negative
eigenvalues of the operator
Hα = H 0 − α V .
(1.59)
P
In [50] (see Section 4.1) we have obtained an estimate on j Fs (|Ej |) in terms
of the potential V . The idea is to compare the eigenvalues of Hα with the
eigenvalues of the two-dimensional Schrödinger operator −∆ − αṼ in L2 (R2 ),
where we define
2
Ṽ (x1 , x2 ) =
d
Zd
V (x1 , x2 , x3 ) sin2
0
πx 3
d
dx3 .
Then, we apply inequality (1.56) to −∆ − αṼ . For the sake of simplicity we
formulate the result again only for symmetric potentials, which means potentials
for which Ṽ (x1 , x2 ) = Ṽ (r) with r2 = x21 + x22 . Hence we get
X
Fs (|Ej |) ≤ c α kṼ ln r/skL1 (B(s)) + α kṼ kL1 (R2 ) + α3/2 kV 3/2 kL1 (Ω) .
j
We point out that this estimate too reflects the correct behaviour in α for both
weak and strong coupling limits. Indeed, for small α the r.h.s. is dominated by
the terms linear in α, while for large α it is the term proportional to α3/2 that
wins over the other two. Here, we again see the differences caused by the mixed
dimensionality of the layer, which is, so to say, locally three-dimensional, but
globally two-dimensional. Analogous estimates for bound states in quantum
waveguides were proved in [37].
Geometrical perturbations (Section 4.2). In [49] we extend the result of
[50] to the bound states induced by a local enlargement of the layer Ω. More
precisely, we study the layer
Ωf := {x1 , x2 , x3 ∈ R3 : 0 < x3 < d + f (x1 , x2 )},
where f : R2 → [0, ∞). We then obtain estimates on the negative eigenvalues
of the operator
π2
Hf = −∆Ωf − 2
in L2 (Ωf )
d
in terms of the deformation function f . In this case, we compare Hf with a
certain two-dimensional operator −∆ − Vf , where the auxiliary potential Vf
depends on f . Without going into the details, we point out that for weakly
deformed layers, i.e. for small f , the leading term in Vf is proportional to
π2
π2
−
,
2
d
(d + f )2
see Section 4.2.
40
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46
Chapter 2
Schrödinger operators in
waveguides
2.1
Waveguides with magnetic field
Published in Comm. Partial Differential Equations 30 (2005) 539–565.
47
Communications in Partial Differential Equations, 30: 539–565, 2005
Copyright © Taylor & Francis, Inc.
ISSN 0360-5302 print/1532-4133 online
DOI: 10.1081/PDE-200050113
Stability of the Magnetic Schrödinger Operator
in a Waveguide
TOMAS EKHOLM1 AND HYNEK KOVAŘÍK2∗
1
Department of Mathematics, Royal Institute of Technology,
Stockholm, Sweden
2
Faculty of Mathematics and Physics, Stuttgart University,
Stuttgart, Germany
The spectrum of the Schrödinger operator in a quantum waveguide is known to
be unstable in two and three dimensions. Any local enlargement of the waveguide
produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is
bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic
field is added into the system. The spectrum of the magnetic Schrödinger operator
is proved to be stable under small local deformations and also under small bending
of the waveguide. The proof includes a magnetic Hardy-type inequality in the
waveguide, which is interesting in its own right.
Keywords Hardy inequality; Magnetic field; Schrödinger operator.
Mathematics Subject Classification 35P05; 81Q10.
1. Introduction
It has been known for a long time that a bending of a two-dimensional quantum
waveguide induces the existence of bound states Exner and Šeba (1989), Goldstone
and Jaffe (1992), Duclos and Exner (1995). From the mathematical point of view,
this means that the Dirichlet Laplacian on a smooth asymptotical straight planar
waveguide has at least one isolated eigenvalue below the threshold of the essential
spectrum. Similar results have been obtained for a locally deformed waveguide,
which corresponds to adding a small “bump” to the straight waveguide; see Bulla
et al. (1997) and Borisov et al. (2001). In both cases an appropriate transformation
Received May 6, 2004; Accepted October 11, 2004
∗
Also on leave of absence from Nuclear Physics Institute, Academy of Sciences, Řež,
near Prague, Czech Republic.
Address correspondence to Tomas Ekholm, Department of Mathematics, Royal
Institute of Technology, S-10041 Stockholm, Sweden; E-mail: [email protected] or Hynek
Kovar̆ík, Stuttgart University, Faculty of Mathematics and Physics, Stuttgart University,
D-750 69 Stuttgart, Germany, Fax: 49-711-6855594; E-mail: [email protected]
539
540
Ekholm and Kovařík
is used to pass to a unitary equivalent operator on the straight waveguide with an
additional potential, which is for small perturbations of the described type proved
to be dominated by an attractive term. As a result, at least one isolated eigenvalue
appears below the essential spectrum for any bending, satisfying certain regularity
properties, respectively for an arbitrarily small “bump.” The crucial point is that for
low energies the Dirichlet Laplacian in a planar waveguide in 2 behaves effectively
as a one-dimensional system, in which the Schrödinger operators with attractive
potentials have a negative discrete eigenvalue no matter how weak the potential is.
This is related to the well-known fact that the Hardy inequality fails to hold in
dimensions one and two.
The purpose of this paper is to prove that in the presence of a suitable magnetic
field some critical strength of the deformation is needed for these bound states to
appear.
The magnetic field is not supposed to affect the essential spectrum of the
Dirichlet Laplacian. We will deal with two generic examples of a magnetic field, a
locally bounded field and an Aharonov–Bohm field. The crucial technical tool of
the present work is a Hardy type inequality for magnetic Dirichlet forms in the
waveguide. For d ≥ 3 the classical Hardy–inequality states that
4
ux2
ux2 dx
dx
≤
x2
d − 22 d
d
(1.1)
for all u ∈ H 1 d . Hence if d ≥ 3 and V ∈ C0 d , V ≥ 0, the operator − −V
does not have negative eigenvalues for small values of the parameter . However, if
d = 1 2 and V ∈ C0 d is nonnegative, then
d
Vxux2 dx ≤
d
ux2 dx
(1.2)
implies V = 0. It is known that if V is nonnegative and not identically zero,
then the spectrum of − −V contains some negative eigenvalues for any > 0;
see Blankenbelcer et al. (1977), (Weidl, 1999, Lemma 5.1). Let us consider the
magnetic Schrödinger operator −i + A2 , where A 2 → 2 is a magnetic vector
potential. Laptev and Weidl (1999) proved a modified version of the inequality (1.1)
in 2 for the quadratic form of a magnetic Schrödinger operator,
Const
2
ux2
−i + Aux2 dx
dx
≤
1 + x2
2
(1.3)
and gave a sharp result for the case of an Aharonov–Bohm field. See Laptev and
Weidl (1999) for details and precise assumptions on the magnetic field under which
inequality (1.3) holds true. This work was later extended in Balinsky (2003), to
multiple Aharonov–Bohm magnetic potentials; see also Evans and Lewis (2004),
Balinsky et al. (in press), and Melgaard et al. (2004). Recently another generalization
of the result by Laptev and Weidl was obtained in Balinsky et al. (2004).
In our model we study the spectrum of −i + A2 in L2 × 0 with
Dirichlet boundary conditions. An essential difference to the above mentioned
cases is that due to the Dirichlet boundary conditions, the spectrum starts from 1.
Consequently, inequality (1.3) becomes trivial. Therefore we shall subtract the
Stability of the Magnetic Schrödinger Operator in a Waveguide
541
threshold of the spectrum and prove a Hardy–inequality in the form
Const
×0
ux2
−i + Aux2 − ux2 dx
dx
≤
2
1 + x1
×0
(1.4)
1
× 0 . Inequality (1.4) is then used
for all u in the magnetic Sobolev space H0A
to prove the stability of the spectrum of the corresponding magnetic Schrödinger
operator under local geometrical perturbations. Stability results in another form of
the magnetic Schrödinger operator on d , d ≥ 2, was also studied in Mantoiu and
Richard (2000).
The text is organized in the following way. The main results are formulated in
Section 2. In Section 3 we prove the Hardy inequalities for the magnetic Schrödinger
operator with a locally bounded field and an Aharonov–Bohm field. The main new
ingredient of our result is that we subtract the threshold of the essential spectrum.
We also prove the asymptotical behavior of the corresponding constant in the
Hardy inequality in the limit of weak fields.
In Section 4 we prove the stability of the essential spectrum of the operator
−i + A2 in locally deformed and curved waveguides for certain magnetic
potentials; see Theorem 4.1.
In Section 5 we use the Hardy inequalities to prove that the spectrum of
−i + A2 is stable under weak enlargement of the waveguide. We also give an
asymptotical estimate on the critical strength of the deformation, for which the
discrete spectrum of −i + A2 will be empty. In particular, if the magnetic field
equals B, then the critical strength of the deformation is proportional to 2 as
→ 0. Moreover, we prove by a trial function argument that the same behavior
of with another constant, is sufficient also for the presence of eigenvalues below
the essential spectrum; see Theorem 5.2. The latter shows that the order of in our
estimate is optimal.
Locally curved waveguides are studied in Section 6. We consider a waveguide
with the curvature , where is a positive parameter and is a fixed smooth
function with compact support. Similarly as in Section 5, we show that there exists
a 0 such that for all < 0 there will be no discrete spectrum of −i + A2 . The
behavior of 0 in the limit of weak fields is at least proportional to 2 , as → 0.
2. The Main Results
Here we formulate the main results of the paper without giving any explicit
estimates of the involved constants. Explicit formulae for these constants are given
in the respective sections.
2.1. Hardy-Type Inequalities
We state a Hardy inequality for magnetic Dirichlet forms separately for the case of
an Aharonov–Bohm field and for a locally bounded field. The following notation
will be used. Let ⊂ 2 be the strip given by = × 0 , ℬr p be the open
ball centred at p with radius r, and
p r =
1 Bx ydx dy
2 ℬr p
(2.1)
542
Ekholm and Kovařík
be the flux of B through the ball ℬr p. We will say that A = a1 a2 is a magnetic
2
vector potential associated with a magnetic field B ∈ L
loc if B = curl A in the
2
1
2
distributional sense and a1 a2 ∈ Lloc . Let H0A denote the completion of
C0 in the norm
u2H 1 = u2L2 + −i + Au2L2 A
(2.2)
2
Theorem 2.1. Let B ∈ L
loc be a real-valued magnetic field. Assume that there is a
ball ℬR p ⊂ , with p = x0 y0 such that the function p r is not identically zero
for r ∈ 0 R. Then
cH
u2
−i + Au2 − u2 dx dy
dx
dy
≤
2
1 + x − x0 (2.3)
1
, where A is a magnetic vector potential associated with B and
holds for all u ∈ H0A
cH is a positive constant given in (3.34).
Theorem 2.2. Assume there is a point p = x0 y0 and a radius R, such that ℬR p ⊂
. Let A be a magnetic vector potential, such that A ∈ L2loc \ℬR p, and for x y ∈
ℬR p we have
−y + y0
x − x0
Ax y = ·
(2.4)
x − x0 2 + y − y0 2 x − x0 2 + y − y0 2
where ∈ \. Then
cAB
u2 dx dy
−i + Au2 − u2 dx dy
≤
2
2
x − x0 + y − y0 (2.5)
1
\p, where cAB is a positive constant given in (3.66).
holds for all u ∈ H0A
Remark. With the translational invariance in mind, we will throughout the paper
assume that x0 = 0.
2.2. Geometrically Deformed Waveguides in a Magnetic Field
As an application of Theorem 2.1 and Theorem 2.2 we prove stability results for the
spectrum of the magnetic Schrödinger operator under geometrical perturbations of
the waveguide. We will consider the following two cases.
2.2.1. Local Enlargement. Let f = 0 be a nonnegative function in C01 , ≥ 0,
and put
= s t ∈ 2 0 < t < + fs (2.6)
Let Md be the self-adjoint operator associated with the closed quadratic form
−i + Au2 ds dt
(2.7)
Stability of the Magnetic Schrödinger Operator in a Waveguide
543
1
, where A is either the magnetic vector potential for the Aharonov–
on H0A
Bohm field, given by
−y + y0
x
Ax y = ·
∈ \
(2.8)
x2 + y − y0 2 x2 + y − y0 2
2
or a magnetic vector potential associated with a magnetic field B ∈ L
0 , such
that B is nontrivial in . Then the following statements hold:
2
Theorem 2.3. Suppose that B ∈ L
0 , such that B is nontrivial in . Then there
is a positive number 0 depending on f , f ′ , a1 and a2 such that for ∈
0 0 the discrete spectrum of Md is empty.
Theorem 2.4. Let A be defined by (2.8). Then there exists a positive number 0
depending on f and f ′ such that for ∈ 0 0 the discrete spectrum of Md is
empty.
2.2.2. Curved Waveguides. Let a and b be real-valued functions in C 2 . Define
the set
= s t s = ax − yb′ x t = bx + ya′ x where x y ∈ (2.9)
where is to be explained below and = × 0 . We assume the normalization
a′ x2 + b′ x2 = 1
(2.10)
for all x ∈ . The boundary of for which y = 0 is a curve ∈ 2 given by
= ax bx x ∈ (2.11)
and the signed curvature → of is given by
x = b′ xa′′ x − a′ xb′′ x
(2.12)
Assume that ∈ C01 and let the natural condition x > − 1 hold for all x ∈ .
We prohibit to be self-intersecting. To be able to study mildly curved waveguides
we consider the waveguide with curvature , where is a positive number and is
a given curvature. For the sake of simplicity we will write instead of .
Let Mc be the self-adjoint operator associated with the closed quadratic form
−i + Au2 ds dt
(2.13)
1
, where A is either the magnetic vector potential for the Aharonov–
on H0A
Bohm field given in (2.8) or a magnetic vector potential associated with a magnetic
2
field B ∈ L
0 , such that B is nontrivial in . Then the following statements
hold:
2
Theorem 2.5. Suppose that B ∈ L
0 , such that B is nontrivial in . There exists
a positive number 0 depending on , ′ , a1 , and a2 such that for ∈
0 0 the discrete spectrum of Mc is empty.
544
Ekholm and Kovařík
Theorem 2.6. Let A be as given in (2.8). There exists a positive number 0 depending
on and ′ such that for ∈ 0 0 the discrete spectrum of Mc is empty.
3. Hardy-Type Inequalities
In this section we will prove Theorems 2.1 and 2.2. Since one would like to know
how the corresponding constants depend on the physical parameters of the model,
we will keep the calculations as explicit as possible, although it makes the proofs
rather technical.
3.1. Proof of Theorem 2.1
The substitution ux y = vx y sin y transforms inequality (2.3) into
cH
v2 sin2 y
−i + Av2 sin2 y dx dy
dx
dy
≤
1 + x − x0 2
(3.1)
for all v ∈ HA1 . This inequality and hence also Theorem 2.1 will be proved in
three steps. First we prove an inequality similar to (3.1) in a ball surrounding a
certain point of the strip, see Lemma 3.1. This result will then be extended to a box
enclosing the given ball; see Lemma 3.2; and finally to the whole strip.
3.1.1. Step 1. Hardy Inequality in a Circle.
Lemma 3.1. Let the magnetic field B and the ball ℬR p be given as in Theorem 2.1.
Then
−i + Au2 dx dy
u2 dx dy ≤ c1
(3.2)
ℬR p
ℬR p
holds for all u ∈ HA1 ℬR p, where A is any magnetic vector potential associated with
B and the constant c1 is given by (3.15).
Proof. We can construct a magnetic vector potential Ax y = a1 x y a2 x y
associated with B this way:
a1 x y = −y − y0 a2 x y = x
0
0
1
Btx ty − y0 + y0 t dt
1
Btx ty − y0 + y0 t dt
(3.3)
(3.4)
Then curl A = x a2 − y a1 = B in the distributional sense and Ax y · x y − y0 =
0 for all x y ∈ 2 . Owing to gauge invariance we can assume that the components
of A are given by (3.3) and (3.4).
Let r be polar coordinates centred at the point p. In these coordinates
inequality (3.2) is equivalent to
2
ur + r −2 − iu + rar u2 r dr d
(3.5)
u2 r dr d ≤ c1
ℬR p
ℬR p
where ar = A · − sin cos .
Stability of the Magnetic Schrödinger Operator in a Waveguide
545
For fixed r we consider the operator Kr = −i + rar in L2 0 2, which
was studied in Laptev and Weidl (1999). The operator Kr is self-adjoint on the
domain H 1 0 2 with
boundary conditions. The spectrum of Kr is discrete,
periodic
and the eigenvalues k k=− and the orthonormal set of eigenfunctions k k=−
are given by
k = k r = k +
r 2
ar d = k + p r
2 0
(3.6)
and
1
k r = √ ei
k −ir 0 ars ds 2
(3.7)
The quadratic form of Kr2 satisfies the inequality
r2
2
0
u2 d ≤
0
2
− iu + rau2 d
(3.8)
for all ur · ∈ H 1 0 2, where r = distp r . Thus
ℬR p
2 2
r −2 − iu + rau2 r dr d
u
r
dr
d
≤
r2
ℬR p
holds for all u ∈ H 1 ℬR p.
Define the function 0 R → 0 1 by
2 r2
r = 0 2 where 0 =
r
r
max
r∈0R r
−1
(3.9)
(3.10)
Since p is piecewise continuous differentiable and p 0 = 0 it follows that is
well-defined. It is clear that r ∈ 0 1 and that there exists at least one r0 ∈
0 R such that r0 = 1. Let v ∈ H 1 0 R such that vr0 = 0; then we have the
inequalities
R
r0
vr2 r dr ≤
2R3 − 3R2 r0 + r03 R ′
v r2 r dr
6r0
r0
(3.11)
r02 r0 ′
v r2 r dr
2
j01
0
(3.12)
and
0
r0
vr2 r dr ≤
where j01 ≥ 2 is the first zero of the Bessel function J0 . Using (3.11) and (3.12), we
conclude that
u2 + 1 − u2 r dr d
u2 r dr d ≤ 2
ℬR p
ℬR p
≤ 202
ℬR p
r −2 − iu + rau2 r dr d
546
Ekholm and Kovařík
+2
≤ 202
0
ℬR p
+ c0
≤ c1
2
r02 r0
1 − u′ 2 r dr
2
j01
0
2R3 − 3R2 r0 + r03 R
′ 2
+
1 − u r dr d
6r0
r0
r −2 − iu + rau2 r dr d
ℬR p
ℬR p
′ u2 + ur 2 r dr d
2
ur + r −2 − iu + rau2 r dr d
(3.13)
where
−2 2
c0 = 4 max j01
r0 6r0 −1 2R3 − 3R2 r0 + r03 c1 = max 202 + 4c0 c22 04 c0 c2 = max r −2 r′ r − r
r∈0R
(3.14)
(3.15)
(3.16)
Remark. Note that the constant c1−1 gives a lower bound of the lowest eigenvalue of
the magnetic Schrödinger operator on the disc of radius R with magnetic Neumann
boundary conditions.
3.1.2. Step 2. Extension to a Box.
Lemma 3.2. Let A, B, and ℬR p be given as in Theorem 2.1. Put R = −R R ×
0 ; then
−i + Au2 sin2 y dx dy
(3.17)
R2 − x2 u2 sin2 y dx dy ≤ c4
R
R
holds for all u ∈ H 1 R , where c4 is a positive constant given in (3.26).
d2
u ∈ H01 0 ∩ H 2 0 uy0 = 0
Proof. The operator − dy
2 − 1 on the domain
is greater than or equal to
c3 = 2 min y0−2 − y0 −2 − 1
(3.18)
In terms of quadratic forms this means that for v in H 1 0 satisfying vy0 = 0 we
have
v′ y2 sin2 y dy
(3.19)
vy2 sin2 y dy ≤ c3−1
0
0
Let u ∈ H 1 R and let 0 → 0 1 be defined by

 √y − y0 if h− x < y < h+ x
2
2
y =
 R −x
1
otherwise.
(3.20)
Stability of the Magnetic Schrödinger Operator in a Waveguide
where h± x = y0 ±
0
√
547
R2 − x2 . We write u = 1 − u + u and use (3.19) to obtain
u2 sin2 y dy ≤ 2
h+ x
h− x
4
+
c3
1 − u2 sin2 y dy
0
2
2
uy sin y dy +
h+ x
h− x
u2 sin2 y dy
(3.21)
R2 − x 2
Hence we get
R
R2 − x2 u2 sin2 y dy dx ≤
2R2 c3 + 4 u2 sin2 y dx dy
c3
ℬR p
4R2 +
u 2 sin2 y dy dx
c 3 R y
(3.22)
for all u ∈ H 1 R . In particular it holds for u, where u ∈ C R , hence
R
R2 − x2 u2 sin2 y dy dx ≤
2R2 c3 + 4 u2 sin2 y dx dy
c3
ℬR p
4R2 +
uy 2 sin2 y dy dx
c 3 R
(3.23)
We can estimate the first term on the r.h.s. by Lemma 3.1 and the second term by
the diamagnetic inequality (see, for instance, Kato, 1973; Simon, 1979; Avron et al.,
1978 or Hundertmark and Simon, 2004) saying that
ux y ≤ −i + Aux y
(3.24)
holds almost everywhere. We get
R
2R2 c1 c3 + 4c1
−i + Au2 sin2 y dx dy
c3 cos2 y0 − 2 + R ℬR p
4R2 +
−i + Au2 sin2 y dx dy
c3 R
= c4
−i + Au2 sin2 y dx dy
(3.25)
R2 − x2 u2 sin2 y dx dy ≤
R
for all u ∈ C R , with
c4 =
2R2 c1 c3 + 4c1 + 4R2
c3 cos2 y0 − 2 + R
The theorem now follows by continuity.
(3.26)
548
Ekholm and Kovařík
3.1.3. Step 3. Extension to the Strip.
Proof of Theorem 2.1. We need the classical one-dimensional Hardy inequality
saying that
−
u2
u′ 2 dt
dt
≤
4
t2
−
(3.27)
holds for any u ∈ H 1 , such that u0 = 0 (see Hardy, 1920). Let m =
define the mapping → 0 1 by

 1
x = x

m
if x > m
R
√
2
and
(3.28)
if x < m
Let u ∈ H 1 . By writing u = u + u1 − and using (3.27) we obtain
u2
u2 sin2 y
2
u1 − 2 sin2 y dx dy (3.29)
dx
dy
≤
2
sin
y
dx
dy
+
2
2
1 + x2
x
≤ 16
u2 sin2 y dx dy
ux 2 + u′ 2 sin2 y dx dy + 2
≤ 16
ux 2 sin2 y dx dy + c5
R
m
R2 − x2 u2 sin2 y dx dy
(3.30)
where
c5 =
64 + 4R2
R4
(3.31)
Equation (3.29) holds in particular for u = v, where v is any function from C with bounded support. Thus by (3.24) and Lemma 3.2 we get
v2 sin2 y
−i + Av2 sin2 y dx dy
dx
dy
≤
c
6
2
1+x
(3.32)
c6 = 16 + c4 c5 (3.33)
where
Since the space of functions from C with bounded support is dense in HA1 ,
we conclude that inequality (3.32) holds for all v ∈ HA1 . In view of (3.1) the
statement of the theorem follows with
cH = c6−1 (3.34)
Stability of the Magnetic Schrödinger Operator in a Waveguide
549
3.2. Weak Magnetic Fields
Here we discuss the behaviour of the constant cH in (2.3) for weak magnetic fields.
Let pB r be the flux of the magnetic field B through the ball ℬr p defined in
(2.1) and define the following constants:
k1 =
−1
max r −1 pB r r∈0R
′
r − pB r
k2 = max r −2 rpB
r∈0R
k4 =
2R2 c3 + 42k12 + 4c0 k14 k22 c3 cos2 y0 − 2 + R
(3.35)
(3.36)
(3.37)
Theorem 3.3. If we replace the magnetic field B in Theorem 2.1 by B, where ∈ ,
then the constant cH in (2.3) satisfies the estimate
cH ≥
1 2
+ 4 k4 c5
(3.38)
for → 0.
Proof. First note that the constants c0 , c3 and c5 are independent of . As → 0 we
get c1 = 2k12 + 4c0 k14 k22 −2 and c2 = k2 . This implies that c4 = k4 −2 + 1 and
therefore (3.38) holds.
3.3. Proof of Theorem 2.2
We will again make use of the substitution ux y = vx y sin y. This transforms
inequality (2.5) into
cAB
u2 sin2 y dx dy
−i + Au2 sin2 y dx dy
≤
2
2
x + y − y0 (3.39)
The proof of Theorem 2.2 will again be done in three steps.
3.3.1. Step 1.
Lemma 3.4. Let the magnetic vector potential A and the ball ℬR p be given as in
Theorem 2.2. Then the inequality
cos2 y0 − 2 + p − x y u2 dx dy
2
2
2
− iu + Au sin y dx dy ≥ x2 + y − y0 2
ℬR p
ℬR p
(3.40)
holds true for all u ∈ C ℬR p such that u = 0 in a neighbourhood of p, where
= min − k
k∈
(3.41)
polar coordinates
centred at the point p and put Dn =
Proof. Let us introduce
r n − 1RN −1 < r < nRN −1 , where N is a natural number. Assume that
550
Ekholm and Kovařík
u ∈ C ℬR p such that u = 0 in a neighbourhood of p. In each Dn we have
− iu + Au2 sin2 y dx dy
Dn
=
Dn
2
ur + r −2 − iu + u2 sin2 y0 + r sin r dr d
nR dr d
≥ cos2 y0 − +
− iu + u2
2
N
r
Dn
(3.42)
To study the form on the r.h.s. of (3.42) we make use of the one-dimensional selfadjoint operator K on L2 0 given by
K = −i + (3.43)
K = u ∈ H 1 0 2 u0 = u2 (3.44)
k = k + (3.45)
defined on the set
The spectrum of K is discrete and its eigenvalues k k∈ and the complete
orthonormal system of eigenfunctions k k∈ are given by
and
1
k = √ ei
k − 2
We can write the function u in the Fourier expansion
ur =
k rk (3.46)
(3.47)
k∈
Then we have
Dn
r −1 − iu + u2 dr d ≥
≥
Dn
2
k k k d dr
r −1 k∈
nRN −1
n−1RN −1
≥ 2
Dn
r −1
k∈
k 2 k2 dr
r −1 u2 dr d
(3.48)
Finally we sum up the inequality over the rings. For any N we have
N
nR dr d
− iu + Au2 sin2 y dx dy ≥
cos2 y0 − +
− iu + u2
2
N
r
ℬR p
D
n
n=1
N
dr d
nR
2
≥
cos2 y0 − +
u2
2
N
r
Dn
n=1
N
R
dr d
≥ 2
u2
cos2 y0 − + r +
2
N
r (3.49)
D
n
n=1
Hence the desired result follows as N → .
Stability of the Magnetic Schrödinger Operator in a Waveguide
551
3.3.2. Step 2.
Lemma 3.5. The inequality
uy2 sin2 y dy
≤
c
u′ y2 sin2 y dy
7
y − y0 2
0
0
(3.50)
holds true for all functions u ∈ H 1 0 such that uy0 = 0, where
c7 =
Proof. It is clear that
42
2
2 − max y0 − y0 2
d2
2 min y0−2 − y0 −2 ≤ − 2 dy
(3.51)
(3.52)
From the Hardy inequality,
1 vy2 dy ′
v y2 dy
≤
4 0
y2
0
(3.53)
for all functions v ∈ H 1 0 such that v0 = 0, where is any positive number, it
follows that
d2
1
≤
−
4y − y0 2
dy2
(3.54)
for v ∈ H 1 0 satisfying vy0 = 0. The estimates (3.52) and (3.54) imply that
d2
1
≤ c7 − 2 − 1 2
y − y0 dy
(3.55)
which in terms of the quadratic form means that
0
vy2 dy
v′ y2 − vy2 dy
≤
c
7
y − y0 2
0
(3.56)
holds for all v ∈ H01 0 such that vy0 = 0. The substitution vy = uy sin y
implies that u ∈ H 1 0 and that uy0 = 0. From (3.56) we get
0
uy2 sin2 y dy
u′ y2 sin2 y dy
≤
c
7
y − y0 2
0
for functions u ∈ H 1 0 such that uy0 = 0.
(3.57)
3.3.3. Step 3.
Proof of Theorem 2.2. Because of density arguments it will be enough to establish
inequality (3.39) for u ∈ C with bounded support such that u = 0 in a
neighborhood of the point p.
552
Ekholm and Kovařík
Assume x ∈ −R R and put h± x = y0 ±
Since ux · ∈ H 1 0 , we have
0
√
R2 − x2 . Let be defined by (3.20).
h+ x u2 sin2 y dy
u2 sin2 y dy
2
′ 2
+
u
sin
y
dy
+
2
≤
2c
u
y
7
2
2
x2 + y − y0 2
h− x x + y − y0 0
4c R2 h+ x u2 sin2 y dy
uy 2 sin2 y dy + 2 + 2 7 2
≤ 4c7
2
2
R −x
h− x x + y − y0 0
(3.58)
by Lemma 3.5. Thus the inequality
0
R2 − x2 u2 sin2 y dy
2
uy 2 sin2 y dy
≤
4c
R
7
x2 + y − y0 2
0
h+ x u2 sin2 y dy
+ 2R2 1 + 2c7 2
2
h− x x + y − y0 (3.59)
holds true. Let R = −R R × 0 . We integrate w.r.t. x to get
R2 − x2 u2 sin2 y dx dy
2
≤
4c
R
uy 2 sin2 y dx dy
7
x2 + y − y0 2
R
u2 sin2 y dx dy
+ 2R2 1 + 2c7 (3.60)
2
2
ℬp R x + y − y0 R
By continuity this inequality can be extended for functions in H 1 \p with
Dirichlet boundary condition at the point p. In particular it holds if u = w, where
w ∈ C with bounded support such that w = 0 in a neighborhood of the point
p. By Lemma 3.4 and (3.24) we have
R
R2 − x2 w2 sin2 y dx dy
−i + Aw2 sin2 y dx dy
≤
c
8
x2 + y − y0 2
R
(3.61)
where
c8 =
Put m =
to get
R
√
2
4R2 2 c7 + 2R2 + 4R2 c7
2 cos2 y0 − 2 + R
(3.62)
and let be given by (3.28). We write w = w + w1 − and use (3.27)
w2 sin2 y dx dy
2
2
w′ 2 sin2 y dx dy
w
sin
y
dx
dy
+
16
≤
16
x
2
2
m
x + y − y0 w2 sin2 y dx dy
+2
2
2
m x + y − y0 w2 sin2 y dx dy
(3.63)
≤ 16 wx 2 sin2 y dx dy + c9
2
2
m x + y − y0 Stability of the Magnetic Schrödinger Operator in a Waveguide
553
2
where c9 = 18 + 32
. Inequality (3.63) can be extended by continuity to functions
R2
from H 1 \p with Dirichlet boundary condition at the point p. Finally, if w =
v, where v ∈ C with bounded support such that v = 0 in a neighbourhood of
the point p, we get by (3.24) that
v2 sin2 y dx dy
−i + Av2 sin2 y dx dy
≤
16
2
2
x + y − y0 v2 sin2 y dx dy
+c9
2
2
m x + y − y0 (3.64)
Using inequality (3.61) we have
v2 sin2 y dx dy
−i + Av2 sin2 y dx dy
≤
c
10
2
2
x + y − y0 where the constant c10 = 16 +
2c8 c9
.
R2
(3.65)
This proves inequality (3.39) with
R2 2 cos2 y0 − 2 + R
cAB = 2 2
8 2R + 2c7 2 + 1 + 2c7 9R2 + 162 where and c7 are given by (3.41) and (3.51), respectively.
(3.66)
4. Stability of the Essential Spectrum
We will prove that under some assumptions on the magnetic vector potential a
magnetic field will not change the essential spectrum of the Laplacian.
Let be a subset of 2 with piecewise continuously differentiable boundary
and let us assume that there is a bounded set 0 ⊂ such that \0 consists up to
translations and rotations of two half strips 1 and 2 . By a half strip we denote the
set 0 × 0 . Let M be the self-adjoint operator associated with the quadratic
form
−i + Au2 dx dy
(4.1)
1
on the domain H0A
, where A will be specified below.
Theorem 4.1. If the magnetic vector potential A = a1 a2 is such that aj ∈ L2loc for j = 1 2 and the functions A and div A exist in 2 and are in L2 2 , then
ess M = 1 (4.2)
Proof. We can without loss of generality assume that 2 = 0 × 0 . To
prove that 1 ⊂ ess M we construct Weyl sequences. Assume that is a nonnegative real number. Let hn n=1 be a singular sequence of real-valued test functions
d2
2
for the operator − dx2 in L at such that supp hn ∈ n and such that
554
Ekholm and Kovařík
hn and h′n are uniformly bounded in n. For instance, let ∈ C0 be a
non-negative function such that L2 = 1 and supp ⊂ −1 1. Let


0



2
2x

n x = nn − 1 − n − 1


2n
−2x



+
nn − 1 n − 1
if x < n or x ≥ n2 nn + 1
if n ≤ x <
2
nn + 1
if
≤ x < n2 2
(4.3)
√
then hn can be chosen as a subsequence of n ∗ x · cos x such that the
functions from the subsequence have disjoint support.
Construct the functions
gn x y = hn x sin y
(4.4)
We will prove that gn is a singular sequence for M at 1 + . Clearly gn ∈ M and
gn 2L2 =
hn x2 sin2 y dx dy =
2
h 2 2 > 0
2 n L (4.5)
for every n. Let u be any function in L2 ; then
u gn L2 =
0
sin y
hn xux ydx dy → 0
n
(4.6)
The latter follows since u· y is in L2 0 for a.e. y ∈ 0 . Finally we must show
that M − + 1gn → 0, as n → . There is a constant c depending on hn and
h′n such that
M − 1 + gn 2L2 = c
n
− h′n − hn 2 dx
+
0
n
A2 + div A2 dx dy → 0
(4.7)
(4.8)
as n → . We have proved that 1 + ∈ ess M for all nonnegative , i.e., 1 ⊂
ess M.
To prove the reverse inclusion, ess M ⊂ 1 , it will be enough to prove
that inf ess M ≥ 1. We study the operator MN being M with additional magnetic
Neumann boundary conditions at the intersections 0 ∩ 1 and 0 ∩ 2 . Then
MN can be written as a direct sum of three operators, M1 ⊕ M0 ⊕ M2 in
L2 1 ⊕ L2 0 ⊕ L2 2 . Since the magnetic vector potential is in L2loc 0 , it
follows from Hundertmark and Simon (2004) and Theorem 2.2 in Avron et al.
(1978) that the spectrum of M0 is discrete. By the maximin principle we have
inf ess M ≥ inf ess MN ≥ min inf M1 inf M2 (4.9)
By the diamagnetic inequality we get
inf Mj ≥ inf −DN = 1
(4.10)
Stability of the Magnetic Schrödinger Operator in a Waveguide
555
for j = 1 2, where −DN denotes the Laplace operator in L2 j with Dirichlet
boundary conditions at ∩ j and Neumann boundary conditions at the
intersection 0 ∩ j . Hence the proof is complete.
Corollary 4.2. Assume that the operator M is associated with a magnetic field B ∈
L
0 , then ess M = 1 .
Proof. We can choose 2 such that Bx y = 0 for all x y ∈ 2 . In this case we
can gauge away the magnetic vector potential in 2 . It is clear that the assumptions
of Theorem 4.1 hold.
5. Local Enlargement of the Waveguide
As already mentioned in Section 2.2, we use the Hardy inequalities in order to
prove the stability of the spectrum of the magnetic Schrödinger operator under small
enlargements of the waveguide. We will consider the case of a bounded compactly
supported magnetic field and the Aharonov–Bohm field separately. Let f , , and
Md be given as in Section 2.2.1.
In Bulla et al. (1997) it was proven that the Friedrich extension of − − 1
defined on C0 had negative eigenvalues for all > 0. For small enough values
of > 0 there is a unique simple negative eigenvalue E
, the function E
is
analytic at = 0, and
E
= −
2
fs ds
2
+ 3 (5.1)
We will show that with a compactly supported magnetic field or an Aharonov–
Bohm field added to the model a certain strength of the perturbation is needed for
those eigenvalues to appear.
5.1. Compactly Supported Fields
We remark that if the magnetic field B has compact support then there exists
a magnetic vector potential A = a1 a2 such that a1 a2 ∈ L 2 ; choose, for
instance, a1 and a2 from (3.3) and (3.4).
Proof of Theorem 2.3. From Corollary 4.2 we know that the essential spectrum of
Md coincides with the half-line 1 . It will be sufficient to prove that Md − 1 is
nonnegative. We denote by d the quadratic form associated with Md , i.e.,
d =
− is + a1 2 + − it + a2 2 ds dt
(5.2)
with d = H01 . Define
U
L2 → L2 0 to be the unitary operator given by
U
x y = 1 + fxx 1 + fxy
(5.3)
(5.4)
556
Ekholm and Kovařík
The operator Md is unitary equivalent to the operator
M
= U
Md U
−1 (5.5)
defined on the set U
Md in L2 0 . The form associated with M
is then given by
= d U
−1 (5.6)
defined on the space = U
d . For convenience let gs = 1 + fs, then
= d U
−1 −is + a1 s tgs− 21 s gs−1 t2
=
2 1
(5.7)
+ −it + a2 s tgs− 2 s gs−1 t ds dt
ig ′ x
2
iyg ′ x
x
y
−
i
=
x
y
+
x
y
+
ã
x
yx
y
x
y
1
2gx
gx
0
2 i
+ −
y x y + ã2 x yx y dx dy
gx
where Ax y = ã1 x y ã2 x y = Ax gxy. Straightforward calculation gives
− ix + ã1 2 + − iy + ã2 2 − y 2
=
0
g′
1 g ′ 2 2 yg ′
−
x + x −
x y + y x 2g
4 g
g
y2 g ′ 2 + 1
yg ′ ã1 + f ã2
2
+
y − y dx dy
y + i
g2
g
−
(5.8)
Let be the quadratic form associated with the Schrödinger operator with the
magnetic vector potential A in the space L2 0 . We have
− 2L2 0 = − 2L2 0 y2 2 f ′ 2 − 2
f − 2 f 2
1 f ′ 2 2
2
−
+
y
g2
4 g
0
f ′
y
f ′
x y + y x −
x + x g
2g
yf ′ ã1 + f ã2
+ i
y − y dx dy
g
−
(5.9)
Without loss of generality we can assume that ≤ 1. Let be the characteristic
function of the support of f . The following lower bound holds true:
− 2L2 0 ≥ − 2L2 0 −
· c11 x 2 + y 2 + c12 2 dx dy
0
(5.10)
Stability of the Magnetic Schrödinger Operator in a Waveguide
557
where the constants are given by
1
+ + a1 f ′ 2
c11 =
f 2
c12 =
1 ′ 2
1
f + f ′ + a1 f ′ + a2 f 4
2
+ 2 + a2 f +
(5.11)
(5.12)
By the pointwise inequality
x 2 + y 2 ≤ 2 − i + A2 + A2 2
(5.13)
and Theorem 2.1 we get
− 2L2 0 ≥
1
− 2
c11 − 2L2 0 2
2
cH
2
− c13 1 + d +
dx dy
2
2
0 1 + x
(5.14)
where
d = max supp f
and
c13 = 21 + a1 2 + a2 2 c11 + c12
(5.15)
and cH is the constant from (2.3). Put
0 =
cH
2c13 1 + d2 then the right-hand side of (5.14) is positive for all ∈ 0 0 .
(5.16)
The following corollary is an immediate consequence of the previous theorem
and Corollary 3.3. It shows the asymptotical behavior of 0 for weak magnetic fields
and the dependence on the distance between the magnetic field and the geometrical
perturbation.
Corollary 5.1. If we replace the magnetic field B by B, where ∈ , then
0 ≥
2
+ 4 2k4 k13 c5 1 + d2 (5.17)
as → 0, where
k13 = lim c13 = f 2 + 2f + 4−1 f ′ 2 + 1 + f ′ →0
(5.18)
and the other constants are given in (3.33), (3.37), and (5.15).
Corollary 5.1 gives us a lower bound on 0 . In the next theorem we will present
an upper bound on 0 , i.e., we will give a relation between and for which bound
states exist. Without loss of generality we assume that includes a small triangle
spanned by the points −s 1 s 1, and 0 1 + with s > 0.
558
Ekholm and Kovařík
Theorem 5.2. Let the magnetic field B be replaced by B, where ∈ , and assume
that
2 ≤
s
+ 2 4A2L2 (5.19)
as → 0, where A is any magnetic vector potential associated with B. Then the operator
Md has at least one eigenvalue below the essential spectrum.
Proof. Define the trial function introduced in Bulla et al. (1997), as follows:


sin ye−s
x−s



y
x y = sin

1 + 1 −



0
x
s
x ≥ s 0 < y < x x < s 0 < y < 1 + 1 −
s
otherwise.
(5.20)
Let · = · L2 . A simple calculation gives
2 2
2
2 s + 3 =
1
−
2
2
(5.21)
for → 0. In order to prove that the discrete spectrum of Md is nonempty, it is
enough to show that the inequality
i + A2
<1
2
(5.22)
is satisfied for certain values of and . By (3.3) and (3.4) it follows that A ∈
L2 . Since = 1, we have
s 2 2
i + A2
2
2 A2
2 A2
≤
+
= 1 − 2
+ 3 +
2
2
2
2
2
(5.23)
Taking into account the fact that
2 = s
1
+s+
2s
2
(5.24)
we get
2 ≤
and the proof is complete.
s
+ 2 4A2
(5.25)
Corollary 5.1 together with Theorem 5.2 shows that the order in the asymptotical
behavior of the constant cH given in Corollary 3.3 is sharp.
Stability of the Magnetic Schrödinger Operator in a Waveguide
559
5.2. The Aharonov–Bohm Field
For simplicity we assume that supp f ⊂ 2 .
Proof of Theorem 2.4. Since div A = 0 and A ∈ L2 1 × 0 , it follows from
Theorem 4.1 that the essential spectrum of Md equals 1 . We will prove that
Md − 1 is nonnegative.
Let U
be the unitary mapping given by (5.3) and (5.4). The operator Md is
unitary equivalent to
M
= U
Md U
−1
(5.26)
defined on the set U
Md in L2 0 . The quadratic form associated with Md is
− is + a1 2 + − it + a2 2 ds dt
(5.27)
d =
1
. Hence the form associated with M
is given by
defined on d = H0A
= d U
−1 (5.28)
defined on the space = U
d .
Put gs = 1 + fs and let be the quadratic form associated with the
Schrödinger operator with the magnetic vector potential A in the space L2 0 ,
where Ax y = ã1 x y ã2 x y = Ax gxy. Without loss of generality we
assume that ≤ 1. Similarly as in arriving to (5.9) we get
− 2L2 0 = − 2L2 0 y2 2 f ′ 2 − 2
f − 2 f 2
1 f ′ 2 2
2
+
y −
g2
4 g
0
y
f ′
f ′
x y + y x −
x + x g
2g
yf ′ ã1 + f ã2
y − y dx dy
+ i
g
· c14 x 2 + y 2
≥ − 2L2 0 − −
0
+ c15 + c16 ã21 + ã22 2 dx dy
(5.29)
where c14 = 2f ′ + 3f + f 2 , c15 = 41 f ′ 2 + 21 f ′ , c16 = f ′ +
f , and is the characteristic function of the support of f . We have
− 2L2 0 ≥ − 2L2 0 2c14 − i + A2 − 2 −
0
2c14 + c15 d2 + 2 2
2
2
+
+ 2c14 + c16 ã1 + ã2 dx dy
x2 + y − y0 2
(5.30)
560
Ekholm and Kovařík
where d = max supp f . We use the pointwise inequality
x · ã21 x y + ã22 x y ≤
42 d2 + 2 2 x2 + y − y0 2 (5.31)
to arrive at
− 2L2 0 ≥ − 2L2 0 − 2
c14 − 2L2 0 c17 d2 + c18
−
2 dx dy
2
2
0 x + y − y0 (5.32)
where c17 = 2c14 + c15 + 42 −2 2c14 + c16 and c18 = 2 2c14 + c15 + 42 2c14 +
c16 . From Theorem 2.2 we conclude
1
− 2
c14 − 2L2 0 − 2L2 0 ≥
2
2
c
+ AB − c17 d2 + c18 dx dy
2
2
2
0 x + y − y0 ≥0
for ∈ 0 0 , where cAB is the constant from (3.66) and
cAB
0 =
2c17 d2 + c18 (5.33)
6. Curved Waveguides
Let , and Mc be defined as in Section 2.2.2. Duclos and Exner (1995) gave a
proof based on ideas of Goldstone and Jaffe (1992) of existence of bound states
below the essential spectrum for the Schrödinger operator − in with Dirichlet
boundary conditions, assuming that = 0, see also Bulla and Renger (1995). The
results are given in terms of the curvature and not of the functions a and b.
These functions a and b can be constructed from uniquely up to rotations and
translations from the identities
x
x
1
ax = a0 +
cos
x2 dx2 dx1 (6.1)
0
bx = b0 +
x
0
sin
0
0
x1
x2 dx2 dx1 (6.2)
Our aim is to prove that if we introduce an appropriate magnetic field into the
system it will make the threshold of the bottom of the essential spectrum stable if
the parameter is small enough.
6.1. Compactly Supported Field
Proof of Theorem 2.5. From Corollary 4.2 we know that the essential spectrum of
Mc is 1 . Again we will show that Mc − 1 is nonnegative.
Stability of the Magnetic Schrödinger Operator in a Waveguide
561
The quadratic form c associated with Mc is given by
c =
− is + a1 2 + − it + a2 2 ds dt
(6.3)
on c = H01 . Define the unitary operator
U L2 → L2 0 (6.4)
as
U x y =
1 + yx ax − yb′ x bx + ya′ x
(6.5)
The operator Mc is unitary equivalent to the operator
M = U Mc U−1
(6.6)
acting on the dense subspace M = U Mc of the Hilbert space L2 0 . We
calculate the quadratic form associated with M . Our change of variables gives
us the Jacobian in the form
b′ x + ya′′ x
a′ x
(6.7)

s = 1 + y−1 a′ xx − b′ x + ya′′ xy (6.8)
s t
=
x y
a′ x − yb′′ x
−b′ x
Hence we have
 = 1 + y−1 b′ x − a′ x − yb′′ x t
x
y
and
= c U−1 2
−i − b′ x + ya′′ x x y
x
y
=
+ ã1 x y 1 + yx
0
1 + yx 2 −ib′ xx + a′ x − yb′′ xy x y
+ + ã2 x y 1 + yx
1 + yx ·1 + yxdx dy
(6.9)
where Ax y = ã1 x y ã2 x y = Aax − yb′ x bx + ya′ x. We continue
without writing arguments of the functions and use the identities a′ a′′ + b′ b′′ = 0
562
Ekholm and Kovařík
and a′′ 2 + b′′ 2 = 2 2 ,
2
ia′ ã1 + b′ ã2 x
−
=
x − x + y 2
1 + y2
1 + y
0
−
i−b′ + ya′′ ã1 + a′ − yb′′ ã2 y − y 1 + y
y′
y + y x + x −
21 + y3
21 + y
2 2 ′ 2
y 2 2
2
2
2
+
dx dy
+
+
ã
+
ã
1
2
41 + y4
41 + y2
−
We write the form q as a perturbation of
− ix + a′ ã1 + b′ ã2 2 + − iy + −b′ ã1 + a′ ã2 2 dx dy
=
(6.10)
(6.11)
0
i.e.,
− 2L2 0 = − 2L2 0 2y + y2 2 2
x 2 − iya′ ã1 + b′ ã2 x − x −
1 + y
0
a′′ ã1 − b′′ ã2
′
′
y − y − iy − b ã1 + a ã2 +
1 + y
y′
x + x +
y + y 3
21 + y
21 + y
2 2 ′ 2
y 2 2
2
−
dx dy
(6.12)
+
41 + y4
41 + y2
+
We can easily arrive at the estimate
− 2L2 0 ≥ − 2L2 0 c19 x 2 + y 2 + c20 2 dx dy
−
(6.13)
0
where is the characteristic function of the support of and
c19 = 2 2 + 21 + a1 + a2 + ′ 2
1
′
+ 3a1 + 3a2 + c20 =
2
2
By using Theorem 2.1 and (5.13) we get
1
2
− 2c19 − 2L2 0 − L2 0 ≥
2
c
2
+ H − c21 1 + d2 dx dy
2
2
0 1 + x
(6.14)
(6.15)
(6.16)
Stability of the Magnetic Schrödinger Operator in a Waveguide
563
c21 = 21 + a1 2 + a2 2 c19 + c20 (6.17)
where
d = max supp and
The right-hand side is positive if ∈ 0 0 , with
0 =
cH
2c21 1 + d2 (6.18)
Hence the discrete spectrum of the operator Mc is empty.
The following corollary gives a lower bound on 0 as a function of .
Corollary 6.1. If we replace the magnetic field B by B, where ∈ , then
0 ≥
2
+ 4 2k4 c5 k21 1 + d2 as → 0, where
k21 = lim c21 = 22 2 + 4 + 2−1 +
→0
(6.19)
3 ′
2
(6.20)
The constants are given in (3.33), (3.37), and (6.17).
6.2. The Aharonov–Bohm Field
For simplicity we assume that supp ⊂ 2 .
Proof of Theorem 2.6. Since div A = 0 and A ∈ L2 1 × 0 , it follows from
Theorem 4.1 that the essential spectrum of Mc equals 1 . It will be enough to
prove that Mc − 1 is nonnegative.
Denote by M the operator U Mc U−1 , where U is defined in (6.4) and (6.5).
Let be the form associated with M defined on the domain = U c .
Following the calculations in (6.9)–(6.12), we get
(6.21)
− 2L2 0 = − 2L2 0 2y + y2 2 2
−
x 2 − iya′ ã1 + b′ ã2 x − x 1 + y
0
a′′ ã1 − b′′ ã2
y − y − iy − b′ ã1 + a′ ã2 +
1 + y
y′
x + x +
y + y 3
21 + y
21 + y
2 2 ′ 2
y 2 2
2
−
dx dy
+
41 + y4
41 + y2
+
Without loss of generality we can assume that ≤ 1. Hence
− 2L2 0 ≥ − 2L2 0 (6.22)
c22 x 2 + y 2 + c23 + c24 ã21 + ã22 2 dx dy
−
0
564
Ekholm and Kovařík
where
c22 = 3 + 2 2 +
′
2
1
+ ′ 2
= 1 + 2 (6.23)
c23 =
(6.24)
c24
(6.25)
By the inequality (5.13), Theorem 2.2, and the fact that
xã21 x y + ã22 x y ≤
d 2 + 2
disty0 0 2 x2 + y − y0 2 where d = max supp , we obtain
1
2
− 2c22 − 2L2 0 − L2 0 ≥
2
2
c
dx dy
× AB − c25
2
2
2
0 x + y − y0 (6.26)
(6.27)
with
c25 = d2 + 2 2c22 + c23 + disty0 0 −2 2c2 2 + c2 4
(6.28)
If we choose
0 =
cAB
2c25
it follows that the right hand side of (6.27) is positive.
(6.29)
Acknowledgments
We would like to thank Timo Weidl for his permanent support and numerous
stimulating discussions throughout the project. Many useful comments and remarks
of Denis I. Borisov and Pavel Exner are also gratefully acknowledged. T.E. has been
partially supported by ESF programme SPECT.
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2.2
Waveguide with magnetic field and combined
boundary conditions
Published in Ann. Henri Poincaré 6 (2005) 327–342.
77
Ann. Henri Poincaré 6 (2005) 1 – 16
c Birkhäuser Verlag, Basel, 2005
1424-0637/05/0201-16 $ 1.50+0.20/0
Annales Henri Poincaré
Spectrum of the Magnetic Schrödinger Operator
in a Waveguide with Combined Boundary Conditions
Denis Borisov, Tomas Ekholm and Hynek Kovařı́k
Abstract. We consider the magnetic Schrödinger operator in a two-dimensional strip.
On the boundary of the strip the Dirichlet boundary condition is imposed except
for a fixed segment (window), where it switches to magnetic Neumann1 . We deal
with a smooth compactly supported field as well as with the Aharonov-Bohm field.
We give an estimate on the maximal length of the window, for which the discrete
spectrum of the considered operator will be empty. In the case of a compactly
supported field we also give a sufficient condition for the presence of eigenvalues
below the essential spectrum.
1 Introduction
The existence of bound states of the Laplace operator in the strip with Dirichlet
boundary conditions and Neumann window was proven in [1] and independently
also in [2]. The so called Neumann window is represented by the segment of the
length 2l of the boundary, on which the Dirichlet condition is changed to Neumann.
A discrete spectrum of the Laplace operator with Neumann window appears for
any nonzero length of the Neumann segment. In particular, for small values of l
the eigenvalue emerges from the continuous spectrum proportionally to l 4 . The
asymptotical estimate for small l were established in [3]. The asymptotics expansion of the emerging eigenvalue for small l was constructed formally in [4], while
the rigorous results were obtained in [5].
On the other hand, the results on the discrete spectrum of a magnetic
Schrödinger operator in waveguide-type domains are scarce. A planar quantum
waveguide with constant magnetic field and a potential well is studied in [6], where
it was proved that if the potential well is purely attractive, then at least one bound
state will appear for any value of the magnetic field. Stability of the bottom of the
spectrum of a magnetic Schrödinger operator was also studied in [7, Sec. 9]
In this work we consider the system, where the discrete spectrum in the
absence of magnetic field appears due to the perturbation of the boundary of the
domain rather than due to the additional potential well. We also assume that
the magnetic field is localized in the sense to be specified below. This assumption
rules out the case of a constant field. As it has been recently shown in [8] the
presence of a suitable magnetic field can prevent the existence of bound states
in the Dirichlet strip with a sufficiently small “bump”. Changing the boundary
1 For
the definition of magnetic Neumann boundary conditions see Section 2, Eq. (2.2)
2
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
conditions to Neumann is however a stronger perturbation in the sense that the
existence of a bound state in a waveguide with the bump added to a certain
segment of the boundary implies the existence of a bound state in a waveguide
with Neumann conditions on the same segment, see [1, Cor. 1.3]. Therefore we
cannot mimic the arguments of [8] in the case of the waveguide with Neumann
window and a different approach is needed.
The main technical tool used in [8] is a modified version of the Hardy inequality for the magnetic Dirichlet quadratic form in the two-dimensional strip. In
the present paper we establish a similar inequality in order to prove the absence
of a discrete spectrum of the magnetic Schrödinger operator in the straight strip
with Neumann window. More exactly speaking, we give sufficient conditions on the
magnetic field and the length of the window, under which the discrete spectrum
is empty. The above mentioned version of Hardy inequality enables us to reduce
the problem to the study of a one-dimensional Laplacian with a purely attractive
potential well of a width 2l and a small but fixed positive potential, see Section 4.2
for the details. We then show that for l small enough such a system has no bound
state. The main profit of our method is that it gives us an explicit estimate on the
critical length of the window, depending on the magnetic field, which guarantees
the absence of discrete spectrum.
It is of course natural to ask whether a sufficiently large Neumann window
will lead to the existence of eigenvalues also in the presence of the magnetic field.
In the case of a smooth and compactly supported field we give an answer to this
question using a minimax-like argument.
The article is organized as follows. In Section 2 we define the mathematical
objects that we work with and describe the problem. We also give the statements
of the main results separately for the case of a compactly supported bounded
magnetic field and for the Aharonov-Bohm field. In Section 3 we show that the
essential spectrum of the Dirichlet Laplacian is not affected by the magnetic field,
neither by the presence of Neumann window. Sufficient conditions for the absence
of the discrete spectrum are proved in Section 4. Finally, the question of presence
of eigenvalues is discussed in Section 5.
2 Statement of the problem and the main results
Let x = (x1 , x2 ) be Cartesian coordinates, Ω be the strip {x : 0 < x2 < π},
and γ be the interval {x : |x1 | < l, x2 = 0}. The rest of the boundary will be
indicated by Γ, i.e. Γ = ∂Ω \ γ. We denote by B = B(x) a real-valued magnetic
field and assume that A is a magnetic vector potential associated with B, i.e.
A = A(x) = (a1 (x), a2 (x)) and B = curl A = ∂x1 a2 − ∂x2 a1 . In what follows
we will consider two main cases of magnetic fields B. The first case is a smooth
compactly supported field. Hereinafter by this we denote the field B belonging
to C 1 (Ω) and vanishing in the neighbourhood of infinity. The second one is the
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
3
Aharonov-Bohm field originated by the potential with components
a1 (x) = −
Φ · (x2 − p2 )
,
(x1 − p1 )2 + (x2 − p2 )2
a2 (x) =
Φ · (x1 − p1 )
,
(x1 − p1 )2 + (x2 − p2 )2
(2.1)
where Φ is a constant and 2πΦ is the flux through the point p = (p1 , p2 ) which is
assumed to be inside the strip Ω. We denote by M0 the operator
2
2
(−i∂x1 + a1 ) + (−i∂x2 + a2 )
on the domain D(M0 ) consisting of all functions u ∈ C ∞ (Ω) vanishing in a neighborhood of Γ and in a neighborhood of infinity and satisfying the boundary condition
(2.2)
(−i∂x2 + a2 )u(x) = 0 on γ.
We will call it magnetic Neumann boundary condition. In the case of AharonovBohm field, the functions u ∈ D(M0 ) are assumed to vanish in a neighbourhood of
the point p. Clearly, the operator M0 is non-negative and symmetric in L2 (Ω) and
therefore it can be extended to a self-adjoint non-negative operator by the method
of Friedrich. In what follows we will denote this extension by M . The main object
of our interest is the spectrum of the operator M .
In order to formulate the main results we need to introduce some auxiliary
notations. By Ω(α, β) we will indicate the subset of Ω given by {x ∈ Ω : α < x1 <
β} and Ω± will be the subsets {x ∈ Ω : x1 > l}, {x ∈ Ω : x1 < −l}, respectively.
The symbol Br (q) denotes a ball of radius r centered at a point q in R2 . The flux
of the field through the ball Br (q) is given by
Z
1
Φq (r) =
B(x) dx.
2π Br (q)
Below we give the summary of the main results of the article.
Theorem 2.1. The essential spectrum of the operator M coincides with [1, +∞).
Theorem 2.2. Assume that the field B is smooth and compactly supported and
(1). There exist two balls BR− (p− ) ⊂ Ω− , BR+ (p+ ) ⊂ Ω+ so that at least one of
the fluxes Φp± (r) is not identically zero for r ∈ [0, R± ];
(2). The inequality
l≤
1
(κ− + κ+ )
12
(2.3)
holds true, where
κ± := min πc± ,
c± are defined in Lemma 4.1.
π
4 ln 2 + π|p±
1|
,
(2.4)
4
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
Then the operator M has empty discrete spectrum.
Theorem 2.3. Assume that the field B is the Aharonov-Bohm one with the potential
given by (2.1) and
(1). The point p is (p1 , p2 ), where p1 < −l;
(2). The inequality
l<
holds true, where
κ
6
π
κ := min πc,
4 ln 2 + π|p1 |
(2.5)
(2.6)
,
c is defined in Lemma 4.2.
Then the operator M has empty discrete spectrum.
The next theorem provides a condition, that guarantees the existence of discrete eigenvalues in the case of a smooth and compactly supported field.
Theorem 2.4. Let the field B be smooth and compactly supported, λ = λ(l) be the
lowest eigenvalue of the Laplacian −∆N ,D in the strip Ω subject to the Dirichlet
condition on Γ and Neumann condition on γ. Assume that the inequality
λ(l) + inf max |A(x)|2 < 1
A
(2.7)
Ω
holds, where infimum is taken over all potentials associated with the field B. Then
the operator M has non-empty discrete spectrum.
Remark 2.5. In the case of a smooth compactly supported field B we did not define
the magnetic potential uniquely. In fact, this is not needed, since the spectrum of
the operator M is invariant under the gauge transformation A →
7 A + ∇ϕ, where
ϕ is a real-valued function. We will employ this property in section 5 to show that
under the hypothesis of this theorem the potential A can be chosen such that |A| is
bounded and of compact support. This will imply that the quantity inf max |A(x)|2
A
Ω
in (2.7) is finite.
Remark 2.6. The constants κ± and κ in Theorems 2.2 and 2.3 giving the estimates
for window length depend on the magnetic field. The constants c± and c in (2.4)
and (2.6) are determined by the rational part of the flux and the distance from the
support of the field to the boundary (see (4.3) and (4.16)). The important role of
the fractional part of the flux is the usual property of the system with magnetic
field (see, for instance, [7, Sec. 10], [9, Sec. 6.4]); this is a case in our work too.
The distance between the magnetic field and the window is taken into account by
π
the presence of the terms 4 ln 2+π|p
in (2.4) and by the similar term in (2.6).
±
|
1
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
5
Throughout the article we will often make use of some notations and it is
convenient to introduce them now. The spectrum of an operator T will be indicated
by σ(T ) while the essential spectrum will be denoted by σess (T ). We will employ
the symbol qT = qT [·, ·] for the sesquilinear form associated with a self-adjoint
operator T and D(qT ) will be the domain of the quadratic form produced by the
sesquilinear form qT . The Hilbert space we will work in is L2 (Ω); we preserve the
notation (·, ·) and k · k for the inner product and norm in this space. In all other
cases the notations of the inner product and norm in a Hilbert space H will be
equipped by a subscript H.
3 Proof of Theorem 2.1
To prove the theorem we will need some auxiliary notations and statements. Let
H be a Hilbert space and S be a positive definite operator in H whose domain
is dense in H. By S1 we indicate the Friedrich’s extension of the operator S and
by S2 another self-adjoint positive definite extension of S. By definition, D(qS2 )
is a Hilbert space endowed with the inner product and the norm originated by
the quadratic form qS2 . Since S1 is the Friedrich’s extension of S it follows that
D(qS1 ) is a subspace of D(qS2 ). Let Q be the orthogonal complement D(qS1 )⊥ in
D(qS2 ) in the inner product qS2 [·, ·].
The proof of the theorem is based on the following lemma proven in [10,
Lemma 3.1].
Lemma 3.1. If each bounded subset of Q (in the norm k · kD(qS2 ) ) is compact in
H, then the operator T := S2−1 − S1−1 is compact in H.
In our case L2 (Ω) plays the role of H and S := (−i∇ + A)2 + 1 with D(S) :=
The Friedrich extension S1 of S is in fact the extension of (−i∇ + A)2 +
1 subject to Dirichlet boundary condition. We know from [8] that σess (S1 ) =
[2, +∞). We set S2 := M + 1; we naturally can treat M + 1 as an extension of
S. If we prove that T := S2−1 − S1−1 is compact, then the essential spectra of
the operators S1 and S2 will coincide by the Weyl theorem (see for instance [11,
Ch. 9, Sec. 1]). We will prove the compactness of T by Lemma 3.1. First we will
establish an auxiliary lemma. By ω we indicate some bounded subdomain of Ω
with infinitely differentiable boundary such that dist (γ, Ω \ ω) > 0. In the case of
Aharonov-Bohm field we also assume that the point p does not belong to ω.
C0∞ (Ω).
Lemma 3.2. For each function u ∈ Q the inequality
kuk ≤ ckukL2 (ω) ,
holds true, where the constant c is independent on u.
Proof. In the proof of the lemma we follow the ideas of the proof of Lemma 3.3
in [10]. The domains D(qS1 ) and D(qS2 ) are completions of C0∞ (Ω) and D(M0 ),
respectively, in norm
k(−i∇ + A) · k2 + k · k2 .
6
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
In the case of compactly supported field we can choose the vector potential A
being from C 1 (Ω) which will make this potential bounded on ω. In the case of
Aharonov-Bohm field the potential is in C 1 (ω) as well since the point p does not
belong to ω by assumption. Therefore, each element v of D(S2 ) belongs to H 1 (ω)
due to the inequality:
kvk2H 1 (ω) = k(−i∇ + A)v − Avk2L2 (ω) + kvk2L2 (ω)
≤ 2 k(−i∇ + A)vk2L2 (ω) + kAvk2L2 (ω) + kvk2L2 (ω)
≤ C k(−i∇ + A)vk2L2 (ω) + kvk2L2 (ω) = C(S2 v, v),
(3.1)
where the constant C is independent on v.
We denote by χ = χ(x) an infinitely differentiable function taking values from
[0, 1] and being equal to one in some neighbourhood of γ, which is a subdomain
of ω, and vanishing outside ω. Since S2 ≥ 1 it follows that
kS2−1 uk ≤ kuk.
(3.2)
Let u ∈ Q. Clearly, (1 − χ)S2−1 u ∈ D(qS1 ) ∩ D(S2 ), thus
S2 (1 − χ)S2−1 u, u = (1 − χ)S2−1 u, u D(q
S2 )
= 0.
Using this equality we deduce
Since
kuk2 = (u, u) − S2 (1 − χ)S2−1 u, u = (S2 χS2−1 u, u).
S2 χS2−1 u = χu − 2 ∇(S2−1 u), ∇χ
R2
(3.3)
− (S2−1 u)∆χ − 2 i (A, ∇χ)R2 S2−1 u
due to (3.1)–(3.3) we have
Z
2
kuk ≤
χ|u|2 dx + ckukL2(ω) kS2−1 ukH 1 (ω)
Ω
q
−1
≤ CkukL2(ω) kuk + (S2 u, u) ≤ CkukL2 (ω) kuk,
where C is independent on u. This proves the lemma.
Let us finish the proof of the Theorem. Given a subset K of Q bounded in the
norm k · kD(qS1 ) , we conclude that it is also bounded in H 1 (ω) due to (3.1). By the
well known theorem on compact embedding of H 1 (ω) in L2 (ω) for each bounded
domain with smooth boundary (see, for instance, [12, Ch. 1, Sec. 6]) we have that
the set K is compact in L2 (ω). Applying now Lemma 3.2, we conclude that K
is compact in L2 (Ω). Hence, the assumption of Lemma 3.1 is satisfied and the
operator T introduced above is compact. The proof of Theorem 2.1 is complete.
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
7
4 Absence of the discrete spectrum
This section is devoted to the proof of Theorems 2.2 and 2.3. By Theorem 2.1
we know that the essential spectrum of the operator M is [1, +∞). Thus, the
equivalent formulation of the absence of the discrete spectrum is the following
inequality
inf σ(M − 1) =
inf
k(−i∇ + A)uk2 − kuk2 ≥ 0.
(4.1)
kuk=1
u∈D(qM )
It will be enough to check the infimum for a k · kD(qM ) -dense subset of D(M ).
Hence
inf σ(M − 1) = inf
k(−i∇ + A)uk2 − kuk2 ≥ 0
(4.2)
kuk=1
u∈D(M0 )
In order to prove this we will need some auxiliary statements which will be established in the next two subsections.
4.1
A Hardy inequality
Here we state a Hardy inequality for the quadratic form of the operator M , which
will be one of the crucial tools in the proofs of Theorems 2.2 and 2.3. Let p =
(p1 , p2 ) ∈ Ω be some point and the number R be such that BR (p) ⊂ Ω. Given a
smooth compactly supported field B, we define the function µ(r) := dist (Φp (r), Z),
where we recall that Φp (r) is the flux of the field B through the ball Br (p). We
introduce the function

1

, if Φp (r) 6≡ 0 as r ∈ [0, R],
(4.3)
c(p, R) = 16 + c1 (R)c2 (p, R)

0,
if Φp (r) ≡ 0 as r ∈ [0, R],
where
64 + 4R2
,
R4
2
2R c3 (p2 )c4 (R) + 4c4 (R) + 4R2
c2 (p, R) =
,
c3 (p2 ) cos2 (|p2 − π2 | + R)
c1 (R) =
c3 (p2 ) = π 2 min{p−2
, (π − p2 )−2 } − 1,
2
µ(r) ′ c4 (p, R) = max ,
r
[0,R] 2
2
c5 (R) = max 2µ0 + 4c5 c6 µ40 , c6 ,
(
)
r02 2R3 − 3R2 r0 + r03
c6 (R) = 4 max
2 ,
j0,1
6r0
(4.4)
8
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
and µ0 and r0 are defined by
µ0 :=
r0
1
=
,
max r−1 µ(r)
µ(r0 )
[0,R]
j0,1 is a smallest positive root of the Bessel function J0 .
It was shown in [8] that the function c(p, R) is well defined. Finally, let us
define

 1, if |x1 | > l,
(4.5)
g(x1 ) = 1
 , if |x1 | ≤ l.
4
Lemma 4.1. Assume that the field B is smooth and compactly supported and the
−
condition (1) of Theorem 2.2 is satisfied for the points p− = (p−
1 , p2 ) and p+ =
+ +
(p1 , p2 ), then
Z
Z
2
ρ(x1 )|u| dx ≤
|(−i∇ + A)u|2 − g(x1 )|u|2 dx,
(4.6)
Ω
Ω
holds for all u ∈ D(M0 ), where

c−


− 2,


 1 + (x1 − p1 )
ρ(x1 ) =





0,
c+
,
2
1 + (x1 − p+
1)
if − ∞ < x1 < p−
1,
+
if p−
1 < x1 < p1 ,
(4.7)
if p+
1 < x1 < +∞,
and the constants c± = c(p± , R± ) are given by (4.3).
Proof. We start the proof from the estimate
Z
Z
|u|2
c−
|(−i∇ + A)u|2 − |u|2 dx,
dx
≤
−
− 1 + (x − p )2
−
1
Ω(−∞,p1 )
Ω(−∞,p1 )
1
(4.8)
which is valid for all u ∈ D(M0 ). The proof of this estimate follows from the
calculations of [8, Sec. 6], where the similar inequality
Z
Z
|u|2
|(−i∇ + A)u|2 − |u|2 dx,
(4.9)
dx
≤
c
− 2
Ω
Ω 1 + (x1 − p1 )
is proved for all u ∈ H01 (Ω) with some constant c. The approach employed in [8,
Sec. 3] can be applied to prove the inequality (4.8). We will not reproduce all the
details of this proof and just note that the only modification needed is to replace
the function ϕ defined in [8, Eq. (3.28)] by

R


if x1 < p−
1
1 − √ ,

2

√
−
R
2(p1 − x1 )
ϕ(x) :=
(4.10)
−
if p−

1 − √ < x1 < p1 ,


R
2



0
elsewhere,
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
9
In the same way the inequality
Z
Z
|u|2
|(−i∇ + A)u|2 − |u|2 dx, (4.11)
dx
≤
c+
+
2
1 + (x1 − p1 )
Ω(p+
Ω(p+
1 ,+∞)
1 ,+∞)
holds for all u ∈ D(M0 ), where c+ = c(p+ , R+ ). We will make use of the diamagnetic inequality (see [13])
|∇|u|(x)| ≤ |(−i∇ + A)u(x)|
(4.12)
which holds pointwise almost everywhere in Ω for each u ∈ D(M0 ). In addition
the trivial inequality
Z π
Z π
|∂x2 u|2 dx2 ≥
g|u|2 dx2
(4.13)
0
0
holds for each fixed x1 and all u ∈ D(M0 ). The diamagnetic inequality (4.12) and
the last estimate lead us to the inequality
Z
Z
Z
|(−i∇ + A)u|2 dx ≥
|∇|u||2 dx ≥
g|u|2 dx,
Ω(α,β)
Ω(α,β)
Ω(α,β)
which is valid for all α < β. Combining now this inequality with (4.8), (4.11) we
arrive at the statement of the lemma.
In the case of the Aharonov-Bohm field the similar statement is true.
Lemma 4.2. Assume that the field is generated by Aharonov-Bohm potential given
by (2.1) and that the condition (1) of the theorem 2.3 is satisfied for the point
p = (p1 , p2 ). Then
Z
Z
ρ(x1 )|u|2 dx ≤
|(−i∇ + A)u|2 − g(x1 )|u|2 dx,
(4.14)
Ω
Ω
holds for all u ∈ D(M0 ), where

c

,
2
ρ(x1 ) = 1 + (x1 − p1 )

0,
−∞ < x1 < p1 ,
(4.15)
p1 < x1 < +∞,
the constant c = c(p, Φ) is given by
c(p, Φ) =
R2 µ2 c3 (p2 ) cos2 (|p2 − π2 | + R)
,
8 2µ2 R2 c3 (p2 ) + (8µ2 + 8 + c3 (p2 ))(9R2 + 16π 2 )
(4.16)
µ := dist {Φ, Z}, c2 (p2 ) is the same as in (4.4).
The proof of this lemma is the same as the one of Lemma 4.8. It is also based
on similar calculations of [8, Sec. 7.1], where the inequality (4.9) was proven for
Aharonov-Bohm field. Here one also needs to replace the function φ in [8, Eq.
(3.28)] by the function ϕ defined in (4.10) with p−
1 = p1 .
10
4.2
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
A one-dimensional model
In this section we will show that the inequality (4.2) holds true if the oned2
2
dimensional Schrödinger operator − dx
2 + V in L (R) with certain potential V
1
is non-negative. We will consider the case of a compactly supported field and the
Aharonov-Bohm field simultaneously.
In view of Lemmas 4.1 and 4.2 we have
1
k(−i∇ + A)uk2 − kuk2 = k(−i∇ + A)uk2 − (g u, u)
2
1
1
+ k(−i∇ + A)uk2 + ((g − 2) u, u)
2
2
1
1
2
≥ k(−i∇ + A)uk + ((ρ + g − 2) u, u) ,
2
2
where g is given by (4.5). Here ρ is determined by (4.7) in the case of a compactly
supported field and by (4.15) in the case of the Aharonov-Bohm field. Thus,
inf
k(−i∇ + A)uk2 − kuk2
kuk=1
u∈D(M0 )
1
2
≥
inf
kuk=1
u∈D(M0 )
k(−i∇ + A)uk2 + ((ρ + g − 2) u, u) .
By the diamagnetic inequality (4.12) we have
inf
kuk=1
u∈D(M0 )
≥
=
k(−i∇ + A)uk2 − kuk2
1
2
inf
kuk=1
u∈D(M0 )
1
2
1
=
2
inf
kuk=1
u∈D(M0 )
inf
kuk=1
u∈D(M0 )
k∇|u|k2 + ((ρ + g − 2) u, u)
k∇uk2 + ((ρ + g − 2) u, u)
Z
Ω
|∂x1 u|2 + |∂x2 u|2 dx
!
+ ((ρ + g − 2) u, u) .
Using now (4.13) we arrive at
inf
kuk=1
u∈D(M0 )
k(−i∇ + A)uk2 − kuk2 ≥
≥
1
2
inf
kuk=1
u∈D(M0 )
k∂x1 uk2 + (ρ u, u) + 2((g − 1) u, u) .
(4.17)
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
11
In order to establish the inequality (4.2) it is therefore enough to show that
Z π Z
|ux1 (x)|2 + ρ(x1 )|u(x)|2 + 2(g(x1 ) − 1)|u(x)|2 dx1 dx2 ≥ 0,
0
R
which is equivalent to the inequality
Z
|v ′ |2 + ρ|v|2 + 2(g − 1)|v|2 dx1 ≥ 0,
(4.18)
R
for all v ∈ C0∞ (R). In other words, to prove Theorems 2.2 and 2.3 it is sufficient
to show that the one-dimensional Schrödinger operator
−
d2
+ ρ + 2(g − 1)
dx21
is non-negative in L2 (R). The proof of this fact is the main subject of the next
section.
4.3
The proofs of Theorems 2.2 and 2.3
As it has been shown in the previous section to prove the absence of the eigenvalues
it is sufficient to check the inequality (4.18). Due to the definition of g it can be
rewritten as
Z
Z
3 l
|v ′ (t)|2 + ρ(t)|v(t)|2 dt ≥
|v(t)|2 dt.
(4.19)
2 −l
R
Let us show that under the assumptions of Theorems 2.2, respectively 2.3 this
inequality holds true. We will show it in detail for the case of compactly supported
field only (i.e. for Theorem 2.2); the case of the Aharonov-Bohm field is similar.
We introduce a function
 π
−

 c−
+ arctan(t − p−
1 ) , t < p1 ,
2
(4.20)
φ− (t) := πc

 −,
t ≥ p−
1.
2
−
′
We remind that c− and p−
1 are given in Lemma 4.1. Clearly, φ− (t) = ρ(t) for t < p1
−
′
and φ− (t) = 0 if t ≥ p1 . Keeping these properties in mind for each t ∈ (−l, l) we
deduce the obvious equality
πc−
v(t) = φ− (t)v(t) =
2
=
Z
Z
t
′
(φ− (s)v(s)) ds
−∞
p−
1
ρ(s)v(s) ds +
−∞
Z
t
−∞
φ− (s)v ′ (s) ds,
12
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
where we also employ the fact that by the assumption of Theorem 2.2 we have
p−
1 < −l. The equality obtained, definition of φ− and Cauchy-Schwarz inequality
give rise to an estimate


2 Z
2
Z p−
t
1
π 2 c2−
|v(t)|2 ≤ 2 
ρ(s)v(s) ds + φ− (s)v ′ (s) ds 
−∞
4
−∞
≤2
Z
≤2
πc−
2
p−
1
ρ(s) ds
−∞
Z
p−
1
2
ρ(s)|v(s)| ds +
−∞
Z
p−
1
2
ρ(s)|v(s)| ds +
−∞
Z
Z
t
−∞
φ2− (s) ds
t
−∞
φ2− (s) ds
Z
Z
t
′
−∞
|v (s)| ds
l
′
−∞
2
2
!
!
(4.21)
|v (s)| ds .
Since the function φ− (t) is constant for t > p−
1 it follows that
Z
t
−∞
φ2− (s) ds
=
Z
p−
1
−∞
= c2−
Z
−
φ2− (s) ds + φ2− (p−
1 )(t − p1 )
0
−∞
π
2
= c2− π ln 2 +
+ arctan(s)
2
ds +
π 2 c2−
(t − p−
1)
4
π 2 c2−
(t − p−
1 ).
4
Substituting the last equality into (4.21) and using the expression for φ− (p−
1 ) (see
(4.20)) we arrive at
Z p−
1
2
ρ(s)|v(s)|2 ds
|v(t)| ≤ 2
πc− −∞
!
Z l
4 ln 2
−
′
2
+
+ (t − p1 )
|v (s)| ds .
π
−∞
2
(4.22)
In the case c− = 0 the fraction c1− in this inequality is understood as +∞, so the
inequality valid for all possible values of c− . Integration (4.22) over (−l, l) and
using the obvious equality
Z
p−
1
−∞
ρ(s)|v(s)|2 ds =
Z
0
−∞
ρ(s)|v(s)|2 ds
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
13
lead us to the estimate
Z 0
2
ρ(s)|v(s)|2 ds
|v(t)| dt ≤ 4l
πc− −∞
−l
!
Z l
4 ln 2
−
′
2
+
|v (s)| ds
− p1
π
−∞
!
Z 0
Z l
4l
2
ρ(s)|v(s)|2 ds +
|v ′ (s)|2 ds ,
≤
κ−
−∞
−∞
Z
l
2
where κ− is given by (2.4). We can rewrite this inequality as
κ−
Z
l
2
−l
|v(t)| dt ≤ 4l 2
Z
0
2
ρ(s)|v(s)| ds +
−∞
Z
l
′
−∞
2
!
|v (s)| ds .
(4.23)
This inequality is valid also in the case of c− = 0. In the same way one can easily
prove similar inequality
κ+
Z
l
−l
Z
|v(t)|2 dt ≤ 4l 2
+∞
ρ(s)|v(s)|2 ds +
Z
+∞
−l
0
|v ′ (s)|2 ds ,
(4.24)
where κ+ is given by (2.4). We sum the inequalities (4.23) and (4.24) to get
(κ− + κ+ )
Z
l
−l
|v(t)|2 dt ≤ 4l 2
Z
+
ρ(s)|v(s)|2 ds +
R
Z
+∞
−l
!
Z
l
−∞
|v ′ (s)|2 ds
|v ′ (s)|2 ds .
This implies that
Z
l
8l
|v(t)| dt ≤
κ
−l
2
Z
2
ρ(s)|v(s)| ds +
Z
R
R
′
2
|v (s)| ds ,
where κ = κ− + κ+ . An immediate consequence of the last inequality is that to
satisfy (4.19) it is sufficient to set
l≤
κ
,
12
which coincides with the inequality (2.3). This completes the proof of Theorem 2.2.
The proof of Theorem 2.3 is similar. One just needs to use the inequality
14
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
(4.23) rewritten in a slightly different way:
Z 0
2
ρ(s)|v(s)|2 ds
|v(t)| dt ≤ 4l
πc− −∞
−l
!
Z l
4 ln 2
−
′
2
+
|v (s)| ds
− p1
π
−∞
!
Z l
Z 0
4l
|v ′ (s)|2 ds ,
ρ(s)|v(s)|2 ds +
≤
κ
−∞
−∞
Z
l
2
with κ given by (2.6). This inequality will immediately imply the estimate (4.19)
if the relation (2.5) is satisfied.
5 Presence of eigenvalues
In this section we will prove Theorem 2.4. We will use the formula
inf σ(M − 1) =
inf
k(−i∇ + A)uk2 − kuk2 .
kuk=1
u∈D(qM )
If we find a test function u ∈ D(qM ) such that
k(−i∇ + A)uk2 − kuk2 < 0
this will prove the presence of the discrete spectrum due to Theorem 2.1. Clearly,
D(qM ) is a subspace of H 1 (Ω) consisting of functions that vanish on Γ. The
eigenfunction ψ of −∆N ,D associated with the lowest eigenvalue λ(l) belongs to
D(qM ). We can choose this eigenfunction being real-valued and normalized in
L2 (Ω). Choosing ψ as a test function we have
k(−i∇ + A)ψk2 = k∇ψk2 + kAψk2 = λ(l) + kAψk2 ≤ λ(l) + max |A|2 .
(5.1)
Ω
Here we used the normalization condition for ψ and an obvious relation λ(l) =
k∇ψk2 . The left hand side of inequality (5.1) is invariant under the gauge transformation of the magnetic potential A. Bearing this fact in mind we take the infimum
in (5.1) over all potentials associated with the field B what leads us to
k(−i∇ + A)ψk2 − kψk2 ≤ λ(l) + inf max |A|2 − 1.
A
Ω
By the assumption the right hand side of the last inequality is less than zero, hence
the theorem is proved.
In conclusion let us show that the second term on the left hand side of (2.7) is
finite. It is sufficient to show that it is finite for some A. Let A be some potential
Vol. 6, 2005
Spectrum of the Magnetic Schrödinger Operator in a Waveguide...
15
associated with B. Since B is smooth and compactly supported, the potential
A can be chosen in C 1 (Ω). Therefore it is bounded on each bounded subset of
Ω. The support of B is a compact set, so there exists number b > 0 such that
B = 0 as x ∈ Ω \ Ω(−b, b), i.e. ∂x2 a2 − ∂x1 a1 = 0 as x ∈ Ω \ Ω(−b, b). Since
both domains Ω(−∞, −b) and Ω(b, +∞) are simply connected, this immediately
implies the existence of functions h− ∈ C 1 (Ω(−∞, −b)), h+ ∈ C 1 (Ω(b, +∞)) such
that ∇h− = A as x ∈ Ω(−∞, −b), ∇h+ = A as x ∈ Ω(b, +∞). We introduce the
function


 h− (x)ζ(x1 ), x1 < −b,
−b ≤ x1 ≤ b,
h(x) = 0,


h+ (x)ζ(x1 ), x1 > b,
where ζ(x1 ) is equal to one as |x1 | > 2b and vanishes as |x1 | ≤ b. By definition h ∈
e := A−∇h leads us to a new vector potential A
e
C 1 (Ω). The gauge transformation A
e is compactly supported
associated with the same field B. Moreover the potential A
e ∈ C 1 (Ω), it follows that max |A|
e 2 is
since ∇h = A if |x1 | is large enough. Since A
finite.
Ω
6 Acknowledgments
D.B. has been supported by DAAD (A/03/01031) and partially supported by
RFBR and the program ”Leading scientific schools” (NSh-1446.2003.1). T.E. has
been supported by ESF Program SPECT. D.B. and T.E. thank the Stuttgart
University, where this work has been done, for the hospitality extended to them.
Authors would like to thank T. Weidl for suggesting them the study of the initial
problem and for numerous stimulating discussions.
References
[1] W. Bulla, F. Gesztesy, W. Renger and B. Simon, Weakly coupled bound states
in quantum waveguides, Proc. Amer. Math. Soc. 125, no. 5, 1487–1495 (1997).
[2] P. Exner, P. Šeba, M. Tater, D. Vaněk, Bound states and scattering in quantum waveguide coupled through a boundary window, J. Math.Phys. 37, no.
10, 4867–4887 (1996).
[3] P. Exner and S. Vugalter, Asymptotic estimates for bound states in quantum
waveguides coupled laterally through a narrow window, Ann. Inst H.Poincaré:
Phys. théor. 65, no. 1, 109–123 (1996).
[4] I.Yu. Popov, Asymptotics of bound states for laterally coupled waveguides,
Rep. Math. Phys. 43, no. 3, 427–437 (1999).
[5] R.R. Gadyl’shin, On regular and singular perturbations of acoustic and quantum waveguides, C.R. Mecanique 332, no. 8, 647–652.
16
D. Borisov, T. Ekholm, H. Kovařı́k
Ann. Henri Poincaré
[6] P. Duclos, P. Exner, B. Meller, Resonances from perturbed symmetry in open
quantum dots. Rep. Math. Phys. 47, no. 2, 253–267 (2001).
[7] T. Weidl, Remarks on virtual bound states for semi-bounded operators,
Comm. in Part. Diff. Eq. 24, no. 1&2, 25–60 (1999).
[8] T. Ekholm and H. Kovařı́k, Stability of the magnetic Schródinger operator in a waveguide, to appear in Comm. in Part. Diff. Eq., Preprint:
arXiv:math-ph/0404069.
[9] H.L. Cycon, R.G. Froese, W. Kirsh, B. Simon, Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs
in Physics. Springer Study Edition. Springer-Verlag, Berlin-New York. 1987.
[10] M.S. Birman, Perturbation of the continuous spectrum of a singular elliptic
operator under a change of the boundary and the boundary condition, Vestnik
Leningradskogo universiteta. 1, 22–55 (1962).
[11] Michail S. Birman and Michail Z. Solomyak, Spectral theory of self-adjoint
operators in Hilbert space, D. Reidel Publishing Company, 1987.
[12] O.A. Ladyzhenskaya, The boundary value problems of mathematical physics.
Applied Mathematical Sciences, v. 49. Springer-Verlag, New York, 1985.
[13] D. Hundertmark and B. Simon, A diamagnetic inequality for semigroup differences, J. Reine Angew. Math. 571, 107–130 (2004).
Denis Borisov
Department of Physics and Mathematics
Bashkir State Pedagogical University
October rev. st., 3a
450000 Ufa, Russia
email : [email protected]
Hynek Kovařı́k
Faculty of Mathematics and Physics
Stuttgart University
Pfaffenwaldring, 57
D-70569 Stuttgart, Germany
email : [email protected]
Communicated by Vincent Rivasseau
submitted 11/05/04, accepted 21/09/04
Tomas Ekholm
Department of Mathematics
Royal Institute of Technology
Lindstedtsvägen, 25
S-100 44 Stockholm, Sweden
email : [email protected]
2.3
A Hardy inequality in twisted waveguides
Published in Arch. Rational Mech. Anal. 188 (2008) 245-264.
95
Arch. Rational Mech. Anal. 188 (2008) 245–264
Digital Object Identifier (DOI) 10.1007/s00205-007-0106-0
A Hardy Inequality in Twisted Waveguides
T. Ekholm, H. Kovařík & D. Krejčiřík
Communicated by G. Friesecke
Abstract
We show that twisting of an infinite straight three-dimensional tube with
non-circular cross-section gives rise to a Hardy-type inequality for the associated
Dirichlet Laplacian. As an application we prove certain stability of the spectrum of
the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that
any local bending, no matter how small, generates eigenvalues below the essential
spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along
a reference curve in an appropriate way. In the present paper we show that for any
other rotation some critical strength of the bending is needed in order to induce a
non-empty discrete spectrum.
1. Introduction
The Dirichlet Laplacian in infinite tubular domains has been intensively studied
as a model for the Hamiltonian of a non-relativistic particle in quantum waveguides;
we refer to [6,13,16] for the physical background and references. Among a variety
of results established so far, let us point out the papers [5,6,9,10,15,17] where the
existence of bound states generated by a local bending of a straight waveguide is
proved. This is an interesting phenomenon for several reasons. From the physical
point of view, one deals with a geometrically induced effect of purely quantum
origin, with important consequences for the transport in curved nanostructures.
Mathematically, the tubes represent a class of quasi-cylindrical domains for which
the spectral results of this type are non-trivial.
More specifically, it has been proved in the references mentioned above that
the Dirichlet Laplacian in non-self-intersecting tubular neighborhoods of the form
246
T. Ekholm, H. Kovařík & D. Krejčiřík
Fig. 1. A locally twisted and bent tube of an elliptical cross-section. Twisting and bending
are demonstrated on the left and right part of the picture, respectively
{x ∈ Rd | dist(x, Γ ) < a},
d ≧ 2,
(1)
where a is a positive number and Γ is an infinite curve of non-trivial curvature
vanishing at infinity, always possesses discrete eigenvalues. On the other hand,
the essential spectrum coincides as a set with the spectrum of the straight tube of
radius a. In other words, the spectrum of the Laplacian is unstable under bending.
The bound states may be generated also by other local deformations of straight
waveguides, for example, by a adding a “bump” [4,2,7].
On the other hand, the first two authors of this paper have shown recently in
[7] (see also [1]) that a presence of an appropriate local magnetic field in a twodimensional waveguide leads to the existence of a Hardy-type inequality for the corresponding Hamiltonian. Consequently, the spectrum of the magnetic Schrödinger
operator becomes stable as a set against sufficiently weak perturbations of the type
considered above.
In this paper we show that in tubes with non-circular cross-sections the same
stability effect can be achieved by a purely geometrical deformation which preserves
the shape of the cross-section: twisting. We restrict to d = 3 and replace the
definition (1) by a tube obtained by translating an arbitrary cross-section along a
reference curve Γ , according to a smooth moving frame of Γ (that is the triad of
the tangent and any two normal vectors perpendicular to each other), see Fig. 1.
We say that the tube is twisted provided that (i) the cross-section is not rotationally
symmetric [cf (8) below] and (ii) the projection of the derivative of one normal
vector of the moving frame to the other one is not zero. The second condition can
be expressed solely in terms of the difference between the second curvature (also
called torsion) of Γ and the derivative of the angle between the normal vectors
of the chosen moving frame and a Frenet frame of Γ [cf (14) below]; the latter
determines certain rotations of the cross-section along the curve. In other words,
twisting and bending may be viewed as two independent deformations of a straight
tube. In order to describe the main results of the paper, we distinguish between two
particular types of twisting.
First, when Γ is a straight line, then of course the curvatures are zero and the
twisting comes only from rotations of a non-circular cross-section along the line.
In this situation, we establish Theorem 3 containing a Hardy-type inequality for the
Dirichlet Laplacian in a straight locally twisted tube. Roughly speaking, this tells
A Hardy Inequality in Twisted Waveguides
247
us that a local twisting stabilizes the transport in straight tubes with non-circular
cross-sections.
Second, when Γ is curved, the torsion is in general non-zero, and we show
that it plays the same role as the twisting due to the rotations of a non-circular
cross-section in the twisted straight case. More specifically, we use Theorem 3 to
establish Theorem 1 saying that the spectrum of the Dirichlet Laplacian in a twisted,
mildly and locally bent tubes coincides with the spectrum of a straight tube, which is
purely essential. This fact has important consequences. For it has been proved in [5]
that any non-trivial curvature vanishing at infinity generates eigenvalues below the
essential spectrum, provided the cross-section is translated along Γ according to the
so-called Tang frame [cf (13) below]. We also refer to [12] for analogous results in
mildly curved tubes. But the choice of the Tang frame for the moving frame giving
rise to the tube means that the rotation of the cross-section compensates the torsion.
Our Theorem 1 shows that this special rotation is the only possible choice for which
the discrete eigenvalues appear for any non-zero curvature of Γ ; any other rotation
of the cross-section will eliminate the discrete eigenvalues if the curvature is not
strong enough. In the curved case, we also establish Theorem 2 extending the result
of Theorem 1 to the case when also the torsion is mild.
After submission of this paper, two other works related to the present topic
appeared. First, Grushin has in [12] a result similar to our Theorem 1; namely,
using a perturbation technique developed in [11], he proves that there are no discrete
eigenvalues in tubes which are simultaneously mildly curved and mildly twisted.
Second, by private communication we learned about the results of Bouchitté,
Mascarenhas and Trabucho, [3], who demonstrate the repulsive effect of twisting in bounded tubes by deriving the asymptotics of the eigenvalues as the thickness
of the tube cross-section goes to zero; in the limit they discover an effective potential which has a positive part if and only if the twisting is present in our setting. We
would like to stress that, apart from the different method we use, the importance of
our results lies in the fact that the non-existence of discrete spectrum follows as a
consequence of a stronger property: the Hardy-type inequality of Theorem 3, and
that we are not restricted to thin tubes.
The organization of the paper is as follows. In the following Section 2, we
present our main results; namely, the Hardy-type inequality (Theorem 3) and the
stability result concerning the spectrum in twisted mildly bent tubes (Theorems 1
and 2). The Hardy-type inequality and its local version (Theorems 3 and 4, respectively) are proved in Section 3. In order to deal with the Laplacian in a twisted bent
tube, we have to develop certain geometric preliminaries; this is done in Section 4.
Theorems 1 and 2 are proved at the end of Section 4. In the Appendix, we state a
sufficient condition which guarantees that a twisted bent tube does not intersect.
The summation convention is adopted throughout the paper and, if not otherwise stated, the range of Latin and Greek indices is assumed to be 1, 2, 3 and
2, 3, respectively. The indices θ and τ are reserved for a function and a vector,
respectively, and are excluded from the summation convention. If U is an open set,
we denote by −∆UD the Dirichlet Laplacian in U , that is the self-adjoint operator
associated in L 2 (U ) with the quadratic form Q UD defined by Q UD [ψ] := U |∇ψ|2 ,
ψ ∈ D(Q UD ) := H01 (U ).
248
T. Ekholm, H. Kovařík & D. Krejčiřík
2. Main results
2.1. Twisted bent tubes
The tubes we consider in the present paper are determined by a reference
curve Γ , a cross-section ω and an angle function θ determining a moving frame
of Γ . We restrict ourselves to curves characterized by their curvature functions.
Let κ1 and κ2 be C 1 -smooth functions on R satisfying
κ1 > 0 on I
and
κ1 , κ2 = 0 on R \ I,
(2)
where I is some fixed bounded open interval. Then there exists a unit-speed (that is
parameterised by arc length) C 3 -smooth curve Γ : R → R3 whose first and second
curvature functions are κ1 and κ2 , respectively; Γ is uniquely determined up to congruent transformations. More precisely, the restriction Γ ↾ I can be reconstructed
from κ1 and κ2 by means of a standard procedure (cf [14, Theorem 1.3.6]), and it
possesses a uniquely determined C 2 -smooth distinguished Frenet frame {e1 , e2 , e3 }.
Since κ1 and κ2 vanish outside I , the complement of Γ ↾ I is formed by two straight
semi-infinite lines, and we can extend the triad {e1 , e2 , e3 } to a global C 2 -smooth
Frenet frame of Γ . The latter, also denoted by {e1 , e2 , e3 }, satisfies the Serret–Frenet
equations (cf [14, Section 1.3]) on Γ :
ėi = Ki j e j ,
i ∈ {1, 2, 3},
where the matrix-valued function (Ki j ) has the skew-symmetric form


0 κ1 0
(Ki j ) = −κ1 0 κ2  .
0 −κ2 0
(3)
(4)
Equation (3) can be viewed as the governing equations defining the global Frenet
frame; therefore, the curve through e1 = Γ˙ . The components e1 , e2 and e3 are
the tangent, normal and binormal vectors of Γ , respectively, and κ2 is sometimes
called the torsion (then κ1 is simply called the curvature).
Given a C01 -smooth function θ on R, we define the matrix valued function
cos θ − sin θ
(5)
(Rθµν ) =
sin θ cos θ
(recall the convention for indices from the end of Introduction). Then the triad
{e1 , Rθ2ν eν , Rθ3ν eν } defines a C 1 -smooth moving frame of Γ having normal vectors
rotated by the angle θ (s) with respect to the Frenet frame at s ∈ R. Later, a stronger
regularity of θ will be required, namely,
θ̈ ∈ L ∞ (R).
(6)
Let ω be a bounded open connected set in R2 and introduce the quantity
a := sup |t|.
t∈ω
(7)
A Hardy Inequality in Twisted Waveguides
We assume that ω is not rotationally invariant with respect to the origin, i.e.,
∃α ∈ (0, 2π ),
tµ Rαµ2 , tµ Rαµ3 | (t2 , t3 ) ∈ ω = ω.
249
(8)
We define a twisted bent tube Ω about Γ as the image
Ω := L(R × ω),
(9)
where L is the mapping from R × ω to R3 defined by
L(s, t) := Γ (s) + tµ Rθµν (s) eν (s).
(10)
We make the natural hypotheses that
a κ1 ∞ < 1
L is injective,
and
(11)
so that Ω has indeed the geometrical meaning of a non-self-intersecting tube;
sufficient conditions ensuring the injectivity of L are derived in the Appendix.
Our object of interest is the Dirichlet Laplacian in the tube, −∆Ω
D . In the simplest
case when the tube is straight (that is I = ∅) and the cross-section ω is not rotated
with respect to a Frenet frame of the reference straight line (that is θ̇ = 0), it is easy
to locate the spectrum:
×ω
(12)
spec (−∆R
D ) = [E 1 , ∞),
where E 1 is the lowest eigenvalue of the Dirichlet Laplacian in ω.
Sufficient conditions for the existence of a discrete spectrum of −∆Ω
D were
recently obtained in [5,12]. In particular, it is known from [5] that if the crosssection ω is rotated appropriately, namely in such a way that
θ̇ = κ2 ,
(13)
then any non-trivial bending (that is I = ∅) generates eigenvalues below E 1 , while
the essential spectrum is unchanged.
As one of the main results of the present paper we show that condition (13)
is necessary for the existence of discrete spectrum in mildly bent tubes with noncircular cross-sections:
Theorem 1. Given C01 -curvature functions (2), a bounded open connected set ω ⊂
R2 satisfying non-symmetricity condition (8) and a C01 -smooth angle function θ
satisfying (6), let Ω be the tube as above satisfying (11). If
κ2 − θ̇ = 0,
then there exists a positive number ε such that
κ1 ∞ + κ̇1 ∞ ≦ ε
Here ε depends on κ2 , θ̇ and ω.
=⇒
spec (−∆Ω
D ) = [E 1 , ∞).
(14)
250
T. Ekholm, H. Kovařík & D. Krejčiřík
An explicit lower bound for the constant ε is given by the estimates made in
Section 4.3 when proving Theorem 1; we also refer to Proposition 1 in the Appendix
for a sufficient conditions ensuring the validity of (11).
Theorem 1 tells us that twisting, induced either by torsion or by a rotation
different from (13), acts against the attractive interaction induced by bending. Its
proof is based on a Hardy-type inequality in straight tubes presented in the following
Section 2.2. The latter provides other variants of Theorem 1, for example, in the
situation when also the torsion is mild:
Theorem 2. Under the hypotheses of Theorem 1, with (14) being replaced by
θ̇ = 0,
(15)
there exists a positive number ε such that
κ1 ∞ + κ̇1 ∞ + κ2 ∞ ≦ ε
=⇒
spec (−∆Ω
D ) = [E 1 , ∞).
Here ε depends on θ̇ , ω and I .
We refer the reader to Section 5 for more comments on Theorems 1 and 2.
2.2. Twisted straight tubes
The proof of Theorems 1 and 2 is based on the fact that a twist of a straight tube
leads to a Hardy-type inequality for the corresponding Dirichlet Laplacian. This is
the central idea of the present paper, which is of independent interest.
By the straight tube we mean the product set R × ω. To any radial vector
t ≡ (t2 , t3 ) ∈ R2 , we associate the normal vector τ (t) := (t3 , −t2 ), introduce the
angular-derivative operator
∂τ := t3 ∂2 − t2 ∂3
(16)
and use the same symbol for the differential expression 1 ⊗ ∂τ on R × ω.
Given a bounded function σ : R → R, we denote by the same letter the function
σ ⊗ 1 on R × ω and consider the self-adjoint operator L σ in L 2 (R × ω) associated
with the Dirichlet quadratic form
lσ [ψ] := ∂1 ψ − σ ∂τ ψ2 + ∂2 ψ2 + ∂3 ψ2 ,
(17)
with ψ ∈ D(lσ ) := H01 (R × ω), where · denotes the norm in L 2 (R × ω). Notice
that the spectrum of L σ does not start below E 1 due to the basic inequality
∇ϕ2L 2 (ω) ≧ E 1 ϕ2L 2 (ω) ,
∀ϕ ∈ H01 (ω).
(18)
The connection between L σ and a twisted straight tube is based on the fact
that for σ = θ̇ , L σ is unitarily equivalent to the Dirichlet Laplacian acting in a
tube given by (9) for the choice Γ (s) = (s, 0, 0), after passing to the coordinates
determined by (10). This can be verified by a straightforward calculation.
If σ = 0, L 0 is just the Dirichlet Laplacian in R×ω, its spectrum is given by (12),
and there is no Hardy inequality associated with the shifted operator L 0 − E 1 . The
latter means that given any multiplication operator V generated by a non-zero,
A Hardy Inequality in Twisted Waveguides
251
non-positive function from C0∞ (R × ω), the operator L 0 − E 1 + V has a negative
eigenvalue. This is also true for non-trivial σ in the case of circular ω centered
in the origin of R2 , since then the angular-derivative term in (17) vanishes for the
test functions of the form ϕ ⊗ J1 on R × ω, where J1 is an eigenfunction of the
Dirichlet Laplacian corresponding to E 1 . However, in all other situations there is
always a Hardy-type inequality:
Theorem 3. Let ω be a bounded open connected subset of R2 satisfying the nonsymmetricity condition (8). Let σ be a compactly supported continuous function
with bounded derivatives and suppose that σ is not identically zero. Then, for all
ψ ∈ H01 (R × ω) and any s0 such that σ (s0 ) = 0, we have
|ψ(s, t)|2
ds dt,
(19)
lσ [ψ] − E 1 ψ2 ≧ ch
2
R×ω 1 + (s − s0 )
where ch is a positive constant independent of ψ but depending on s0 , σ and ω.
It is possible to find an explicit lower bound for the constant ch ; we give an
estimate in (27). The particular kind of Hardy weight in (19) is due to the classical
one-dimensional Hardy inequality (26) used in the proof of Theorem 3.
The assumption that σ has a compact support ensures that the essential spectrum
of L σ coincides with (12). As a consequence of the Hardy-type inequality (19), we
get that the presence of a non-trivial σ in (17) represents a repulsive interaction in
the sense that there is no other spectrum for all small potential-type perturbations
having O(s −2 ) decay at infinity.
As explained above, the special choice σ = θ̇ leads to a direct geometric interpretation of L σ in connection with the twisted straight tubes. As another application
of Theorem 3, we shall apply it to the twisted bent tubes, namely, with the choice
σ = κ2 − θ̇ to prove Theorem 1 and with σ = θ̇ to prove Theorem 2 (cf Section 4.3).
Here the main idea is to pass to the curvilinear coordinates induced by (10) in which
the Laplacian −∆Ω
D becomes L σ plus an explicit (second-order) perturbation. The
quadratic form of the perturbation is not of definite sign but it vanishes either if
the C 1 -norm of κ1 tends to zero for σ = κ2 − θ̇ or if both the C 1 -norm of κ1 and
the supremum norm of κ2 tend to zero for σ = θ̇ . Hence the proofs of Theorems 1
and 2 reduce to an algebraic comparison of quadratic forms, the main trouble being
the second order of the perturbation.
3. Hardy inequality for twisted straight tubes
In this section, we establish Theorem 3 in two steps. After certain preliminaries,
we first derive a “local” Hardy inequality (Theorem 4). Then the local result is
“smeared out” by means of a classical one-dimensional Hardy inequality.
3.1. Preliminaries
Definition 1. To any ω ⊂ R2 , we associate the number
λ := inf
∇ϕ2L 2 (ω) − E 1 ϕ2L 2 (ω) + ∂τ ϕ2L 2 (ω)
ϕ2L 2 (ω)
,
252
T. Ekholm, H. Kovařík & D. Krejčiřík
where the infimum is taken over all non-zero functions from H01 (ω).
It is clear from (18) that λ is a non-negative quantity. Our Hardy inequality is based
on the fact that λ is always positive for non-circular cross-sections.
Lemma 1. If ω satisfies (8), then λ > 0.
Proof. The quadratic form b defined on L 2 (ω) by
b[ϕ] := ∇ϕ2L 2 (ω) − E 1 ϕ2L 2 (ω) + ∂τ ϕ2L 2 (ω) , ϕ ∈ D(b) := H01 (ω),
is non-negative [cf (18)], densely defined and closed; the last two statements follow
from the boundedness of τ and from the fact that they hold true for the quadratic form
defining the Dirichlet Laplacian in ω. Consequently, b gives rise to a self-adjoint
operator B. Moreover, since B ≧ −∆ωD − E 1 , and the spectrum of −∆ωD is purely
discrete, the minimax principle implies that B has a purely discrete spectrum, too.
λ is clearly the lowest eigenvalue of B. Assume that λ = 0. Then, firstly, the ground
state ϕ of B and −∆ωD coincide, hence ϕ is analytic and positive in ω; secondly, we
have ∂τ ϕ = 0. This implies that the angular derivative of ϕ is zero. Together with
our assumption on ω we can conclude that there is a point in ω where ϕ vanishes.
This contradicts the positivity of ϕ. ⊓
⊔
Next we need a specific lower bound for the spectrum of the Schrödinger
operator in a bounded one-dimensional interval with Neumann boundary conditions and a characteristic function of a subinterval as the potential.
Lemma 2. Let Λ be a bounded open interval of R. Then for any open subinterval
Λ′ ⊂ Λ and any f ∈ H1 (Λ), the following inequality holds:
f 2L 2 (Λ) ≦ c Λ, Λ′ f 2L 2 (Λ′ ) + f ′ 2L 2 (Λ) ,
where c(Λ, Λ′ ) := max 2 + 16 (|Λ|/|Λ′ |)2 , 4 |Λ|2 .
Proof. Without loss of generality, we may suppose that Λ′ := (−b/2, b/2) with
some positive b. Define a function g on Λ by
2 |x|/b for |x| ≦ b/2,
g(x) :=
1
otherwise.
Let f be any function from H1 (Λ). Then ( f g)(0) = 0 and the Cauchy–Schwarz
inequality gives
x
2
| f (x)g(x)| ≦ |x|
|( f g)′ |2 ≦ |Λ| ( f g)′ 2L 2 (Λ)
(20)
0
for any x ∈ Λ. Now we write f = f g + f (1 − g) to get
f 2L 2 (Λ) ≦ 2 f g2L 2 (Λ) + 2 f (1 − g)2L 2 (Λ) = 2 f g2L 2 (Λ) + 2 f 2L 2 (Λ′ ) .
Using the estimate (20) and the fact that |g ′ | = 2 |Λ′ |−1 on Λ′ , we obtain the
statement of the lemma. ⊓
⊔
A Hardy Inequality in Twisted Waveguides
253
3.2. A local Hardy inequality
Since σ is continuous and has compact support there are closed intervals A j
such that
supp (σ ) =
A j and |Ai ∩ A j | = 0, i = j,
j∈K
where K ⊆ N is an index set. The main result of this subsection is the following
local type of Hardy inequality:
Theorem 4. Let the assumptions of Theorem 3 hold. For every j ∈ K there is a
positive constant a j depending on σ ↾ A j such that for all ψ ∈ H01 (R × ω),
|∂2 ψ|2 + |∂3 ψ|2 + |∂1 ψ − σ ∂τ ψ|2 − E 1 |ψ|2 ≧ a j λ
|σ ψ|2 ,
A j ×ω
A j ×ω
(21)
where λ is the positive constant from Definition 1 depending only on the geometry
of ω.
To prove Theorem 4, it will be useful to introduce the following quantities:
Definition 2. For any M ⊆ R and ψ ∈ H01 (R × ω), we define
I1M := χ M ∇ ′ ψ2 − E 1 χ M ψ2 ,
I2M := χ M ∂1 ψ2 ,
I3M := χ M σ ∂τ ψ2 ,
M
I2,3
:= −2 ℜ (∂1 ψ, χ M σ ∂τ ψ),
where χ M denotes the characteristic function of the set M × ω, ∇ ′ denotes the gradient operator in the “transverse” coordinates (t2 , t3 ) and (·, ·) is the inner product
generated by · .
Note that I1M is non-negative due to (18) and that we have
supp(σ )
supp(σ )
lσ [ψ] − E 1 ψ2 = I1R + I2R + I3
+ I2,3
.
(22)
Let A be the union of any (finite or infinite) sub-collection of the intervals A j .
A .
The following lemma enables us to estimate the mixed term I2,3
Lemma 3. Let the assumptions of Theorem 3 be satisfied. Then for each positive
numbers α and β, there exists a constant γα,β depending also on σ ↾ A such that
for any ψ ∈ H01 (R × ω),
A
|I2,3
| ≦ γα,β I1A + α I2B + β I3A ,
where B := (inf A, sup A).
Proof. It suffices to prove the result for real-valued functions ψ from the dense
subspace C0∞ (R × ω). We employ the decomposition
ψ(s, t) = J1 (t) φ(s, t),
(s, t) ∈ R × ω,
(23)
T. Ekholm, H. Kovařík & D. Krejčiřík
254
where J1 is a positive eigenfunction of the Dirichlet Laplacian on L 2 (ω) corresponding to E 1 (we shall denote by the same symbol the function 1 ⊗ J1 on R × ω),
and φ is a real-valued function from C0∞ (R × ω), actually introduced by (23). Then
I1A = χ A J1 ∇ ′ φ2 ,
I2A = χ A J1 ∂1 φ2 ,
I3A = χ A σ (J1 ∂τ φ + φ ∂τ J1 )2 ,
A
I2,3
= −2 (J1 ∂1 φ, χ A σ (J1 ∂τ φ + φ ∂τ J1 )) ,
where we have integrated by parts to establish the identity for I1A . Using
|σ ∂τ φ|2 ≦ c1 |∇ ′ φ|2 ,
with c1 := σ ↾ A2∞ a 2 ,
and applying the Cauchy–Schwarz inequality and the Cauchy inequality with
A can be estimated as follows:
α > 0, the first term in the sum of I2,3
√
2 c1 A α A
|2 (J1 ∂1 φ, χ A σ J1 ∂τ φ)| ≦ 2 c1 I1A I2A ≦
I + I2 .
(24)
α 1
2
In order to estimate the second term, we first combine integrations by parts to get
|2 (J1 ∂1 φ, χ A σ φ ∂τ J1 )| = φ, χ A σ̇ J1 2 ∂τ φ .
Using
|σ̇ ∂τ φ|2 ≦ c2 |∇ ′ φ|2 ,
with c2 := σ̇ ↾ A2∞ a 2 ,
and the Cauchy–Schwarz inequality, we have
2
φ, χ A σ̇ J1 2 ∂τ φ ≦ c2 I1A χ A J1 φ2 ,
Obviously, we can find an open interval A′ ⊂ A such that there exists a certain
positive number σ0 , for which
σ (s) ≧ σ0 ,
∀ s ∈ A′ .
Lemma 2 tells us that
χ A J1 φ2 ≦ χ B J1 φ2 ≦ c(B, A′ ) I2B + χ A′ J1 φ2
≦ c(B, A′ ) I2B + σ0−2 χ A′ σ J1 φ2 .
Moreover, for each fixed value of s ∈ R we have σ (s) J1 φ(s, ·) ∈ H01 (ω); therefore, we can apply Lemma 1 to obtain
χ A′ σ J1 φ2 ≦ χ A σ J1 φ2 ≦ λ−1 I3A + σ 2∞ I1A .
Writing c3 := c2 c(B, A′ )λ−1 σ0−2 , we conclude that
2
φ, χ A σ̇ J1 2 ∂τ φ ≦ c3 I1A σ 2∞ I1A + λ σ02 I2B + I3A
2
α
≦ γ̃α,β I1A + I2B + β I3A
2
(25)
A Hardy Inequality in Twisted Waveguides
255
√
for any β > 0 and γ̃α,β := max{ c3 σ ∞ , c3 (2β)−1 , c3 λ σ02 α −1 }. Finally,
A | follows by setting γ
combining (24) with (25), the estimate for |I2,3
α,β := γ̃α,β +
−1
2 c1 α . ⊓
⊔
Now we are in a position to establish Theorem 4.
Proof of Theorem 4. We take A = A j , α = 1, β < 1 and keep in mind that γ1,β
in Lemma 3 depends on j. We define γ (β, j) := max{1/2, γ1,β }. Lemma 3 then
gives
|∇ ′ ψ|2 + |∂1 ψ − σ ∂τ ψ|2 − E 1 |ψ|2
A j ×ω
1−β
1
1 Aj
A
A
A
A
I j.
I2 j + I3 j − |I2,3j | +
≧ I1 + 1 −
2
2γ (β, j)
2γ (β, j) 3
A
A
A
Since I2 j + I3 j − |I2,3j | ≧ 0, we get from Lemma 1 that
|∇ ′ ψ|2 + |∂1 ψ − σ ∂τ ψ|2 − E 1 |ψ|2
A j ×ω
≧ aj
where
⊓
⊔
A
A
σ ↾ A j 2∞ I1 j + I3 j ≧ a j λ
A j ×ω
|σ ψ|2 ,
1
1−β
1
.
,
a j = min
2
σ ↾ A j 2∞ γ (β, j)
Remark 1. Note that the Hardy weight on the right-hand side of (21) cannot be
made arbitrarily large by increasing σ , since the constant a j is proportional to
σ ↾ A j −2
∞ if the latter is large enough. We want to point out that this degree
of decay of a j is optimal if the axes of rotation intersects ω. Assume there exists
an α < 2, such that a j is proportional to σ ↾ A j −α
∞ when σ ↾ A j ∞ → ∞.
Consider a test function ψ of the form ψ(s, t) := g(s) f (t), where g ∈ H1 (R)
is supported inside A j and f ∈ H01 (ω) is radially symmetric with respect to the
intersection of ω with the axes of rotation. Then ∂τ ψ = 0 on A j × ω; therefore,
the left-hand side of (21) is for this test function independent of σ . Take σ = n σ̃
with σ̃ being a fixed function. The right-hand side of (21) then tends to infinity as
n → ∞, which contradicts the inequality.
3.3. Proof of Theorem 3
For applications, it is convenient to replace the Hardy inequality of Theorem 4
with a compactly supported Hardy weight by a global one. To do so, we recall the
following version of the one-dimensional Hardy inequality:
|v(x)|2
dx
≦
4
|v ′ (x)|2 dx
(26)
2
x
R
R
T. Ekholm, H. Kovařík & D. Krejčiřík
256
for all v ∈ C0∞ (R) with v(0) = 0. Inequality (26) extends by continuity to all
v ∈ H1 (R) with v(0) = 0.
Without loss of generality we can assume that s0 = 0. Let J = [−b, b], with
some positive number b, be an interval where |σ | > 0. Let f˜ : R → R be defined
by
1
for |s| ≧ b,
f˜(s) :=
|s|/b for |s| < b,
and put f := f˜ ⊗ 1 on R × ω. For any ψ ∈ C0∞ (R × ω), let us write ψ =
f ψ + (1 − f )ψ. Applying (26) to the function s → ( f ψ)(s, t) with t fixed, we
arrive at
R×ω
|ψ(s, t)|2
| f˜(s)ψ(s, t)|2
ds dt ≦ 2
ds dt + 2
|(1 − f )ψ|2
1 + s2
s2
R×ω
J ×ω
≦ 16 (∂1 f )ψ2 + 16 f ∂1 ψ2 + 2 χ J (1 − f )ψ2
16
≦
+ 2 χ J ψ2 + 16 ∂1 ψ2 ,
b2
where χ J denotes the characteristic function of the set J × ω. Theorem 4 then
implies that there exists a positive constant c0 depending on σ such that
−1 lσ [ψ] − E 1 ψ2 .
χ J ψ2 ≦ c0 λ min |σ |
J
To estimate the second term we let A = supp (σ ) and rewrite the inequality of
Lemma 3 for β = 1 as
A
γα−1 |I2,3
| ≦ I1A + α γα−1 I2B + γα−1 I3A ,
where γα := max{1, γα,1 } and α ∈ (0, 1). Substituting this inequality into (22),
A = γ −1 I A + (1 − γ −1 ) I A and employing I A + I A + I A ≧ 0, we
writing I2,3
α
α
2,3
2,3
2
3
2,3
obtain
I2B = χ B ∂1 ψ2 ≦ γα (1 − α)−1 lσ [ψ] − E 1 ψ2 .
On the complement of B × ω, we have a trivial estimate
χR\B ∂1 ψ2 ≦ lσ [ψ] − E 1 ψ2 .
Summing up, the density of C0∞ (R × ω) in H01 (R × ω) implies Theorem 3 with
ch ≧
16 + 2 b2
+ 16
b2 c0 λ min J |σ |2
−1
γα
.
+1
1−α
(27)
A Hardy Inequality in Twisted Waveguides
257
4. Twisted bent tubes
Here we develop a geometric background to study the Laplacian in bent and
twisted tubes, and transform the former into a unitarily equivalent Schrödinger-type
operator in a straight tube. At the end of this section, we also perform proofs of
Theorems 1 and 2 using Theorem 3. We refer to Section 2.1 for definitions of basic
geometric objects used throughout the paper.
While we are mainly interested in the curves determined by curvature functions
of type (2), we stress that the formulae of Sections 4.1 and 4.2 are valid for arbitrary
curves (it is only important to assume the existence of an appropriate Frenet frame
for the reference curve of the tube, cf [5]).
4.1. Metric tensor
Assuming (11) and using the inverse function theorem, we see that the mapping L introduced in (10) induces a C 1 -smooth diffeomorphism between the straight
tube R × ω and the image Ω. This enables us to identify Ω with the Riemannian
manifold (R × ω, G i j ), where (G i j ) is the metric tensor induced by the embedding L, that is
G i j := (∂i L) · (∂ j L),
with the dot being the scalar product in R3 .
Using (3) and the orthogonality conditions Rθµρ Rθνρ = δµν , we find

 2
h + hµhµ h2 h3
(G i j ) = 
h2
1 0 ,
h3
0 1
(28)
where
h(s, t) := 1 − [t2 cos θ (s) + t3 sin θ (s)] κ1 (s),
h 2 (s, t) := −t3 [κ2 (s) − θ̇(s)],
h 3 (s, t) := t2 [κ2 (s) − θ̇ (s)].
Furthermore,
G := det(G i j ) = h 2 ,
which defines the volume element of (R × ω, G i j ) by setting
d vol := h(s, t) ds dt.
Here and in the sequel dt ≡ dt2 dt3 denotes the two-dimensional Lebesgue measure
in ω.
The metric is uniformly bounded and elliptic in view of the first of the assumptions in (11); in particular, (7) yields
0 < 1 − a κ1 ∞ ≦ h ≦ 1 + a κ1 ∞ < ∞.
(29)
T. Ekholm, H. Kovařík & D. Krejčiřík
258
It can be directly checked that the inverse (G i j ) of the metric tensor (28) is
given by


1
−h 2
−h 3
1
(30)
(G i j ) = 2  −h 2 h 2 + h 22 h 2 h 3 .
h
−h 3 h 3 h 2 h 2 + h 23
It is worth noticing that one has the decomposition
(G i j ) = diag(0, 1, 1) + (S i j ),
(31)
where the matrix (S i j ) is positive semi-definite.
4.2. The Laplacian
Recalling the diffeomorphism between R × ω and Ω given by L, we identify the Hilbert space L 2 (Ω) with L 2 (R × ω, d vol). Furthermore, the Dirichlet
Laplacian −∆Ω
D is unitarily equivalent to the self-adjoint operator Q̃ associated on
L 2 (R × ω, d vol) with the quadratic form
q̃[ψ] :=
(∂i ψ) G i j (∂ j ψ) d vol,
R×ω
ψ ∈ D(q̃) := H01 (R × ω, d vol). (32)
We can write Q̃ = −G −1/2 ∂i G 1/2 G i j ∂ j in the form sense, which is a general
expression for the Laplace–Beltrami operator on a manifold equipped with a metric (G i j ).
Now we transform Q̃ into a unitarily equivalent operator Q acting in the Hilbert
space L 2 (R × ω), without the additional weight G 1/2 in the measure of integration.
This is achieved by means of the unitary operator
U : L 2 (R × ω, d vol) → L 2 (R × ω) : ψ → G 1/4 ψ .
Defining Q := U Q̃ U −1 , it is clear that Q is the operator associated with the
quadratic form
q[ψ] := q̃[G −1/4 ψ],
ψ ∈ D(q) := H01 (R × ω).
It is straightforward to check that
q[ψ] = ∂i ψ, G i j ∂ j ψ + ψ, (∂i F)G i j (∂ j F) ψ + 2 ℜ ∂i ψ, G i j (∂ j F) ψ ,
(33)
where
F := log(G 1/4 ).
A Hardy Inequality in Twisted Waveguides
259
4.3. Proof of Theorems 1 and 2
Let us first prove Theorem 1. Putting σ := κ2 − θ̇, we observe that lσ is equal
to q after letting k := κ1 ∞ + κ̇1 ∞ equal to zero in the latter form. Hence, the
proof of Theorem 1 reduces to a comparison of these quadratic forms and the usage
ij
of Theorem 3. Let (G 0 ) be the matrix (30) after letting κ1 = 0, that is with h being
ij
replaced by 1 while h 2 and h 3 being unchanged; then lσ [ψ] = (∂i ψ, G 0 ∂ j ψ).
We strengthen the first of the hypotheses (11) to
κ1 ∞ ≦ 1/(2a),
so that we have a uniform positive lower bound to h, namely h ≧ 1/2 due to (29).
It is straightforward to check that we have on R × ω the following pointwise
inequalities:
ij
max
i, j∈{1,2,3}
|G i j − G 0 | ≦ C1 k χ I ,
max |∂i F| ≦ C2 k χ I ,
i∈{1,2,3}
where χ I denotes the characteristic function of the set I × ω and
2
C1 := 6 a 1 + a κ2 − θ̇ ∞ ,
At the same time,
C2 := 1 + a 1 + θ̇ ∞ .
ij
C3−1 1 ≦ (G 0 ) ≦ C3 1,
in the matrix-inequality sense on R × ω, where 1 denotes the identity matrix and
C3 := 1 + a κ2 − θ̇ ∞ + a 2 κ2 − θ̇2∞ .
Consequently, we have the following matrix inequality on R × ω:
ij
ij
(1 − C4 k χ I )(G 0 ) ≦ (G i j ) ≦ (1 + C4 k χ I )(G 0 ),
where C4 := 3 C1 C3 . Finally, if we assume that k ≦ 1 we have
|(∂i F)G i j (∂ j F)| ≦ C52 k 2 χ I ,
(34)
√
where C5 := C2 3 C3 (1 + C4 ).
Let ψ be any function from H01 (R × ω). First we estimate the term of indefinite
sign on the right-hand side of (33) as follows:
1/2
, χ I |ψ|
2 ℜ ∂i ψ, G i j (∂ j F) ψ ≦ 2 C5 k χ I (∂i ψ)G i j (∂ j ψ)
≦ C52 k χ I ψ2 + k ∂i ψ, χ I G i j ∂ j ψ .
Here the first inequality is established by applying the Cauchy–Schwarz inequality to the inner product induced by (G i ) and using (34). The second inequality
260
T. Ekholm, H. Kovařík & D. Krejčiřík
follows by the Cauchy–Schwarz inequality in the Hilbert space L 2 (R × ω) and by
an elementary Cauchy inequality. Consequently,
ij
q[ψ] ≧ ∂i ψ, (1 − C6 k χ I ) G 0 ∂ j ψ − C7 k χ I ψ2 ,
(35)
where C6 := 1 + C4 and C7 := 2 C52 .
Assume k < C6−1 , using the decomposition of the type (31) for the
ij
matrix (G 0 ), neglecting the positive contribution coming from the correspondij
ing matrix
√ (S0 ), using the Fubini theorem and applying (18) to the function
ϕ := R 1 − C6 k χ I (s) ψ(s, ·) ds, we may estimate (35) as follows:
q[ψ] − E 1 ψ2 ≧ (1 − C6 k) lσ [ψ] − E 1 ψ2 − (C6 E 1 + C7 ) k χ I ψ2 .
Applying Theorem 3 to the right-hand side of the previous inequality, we have
ch (1 − C6 k)
2
− (C6 E 1 + C7 ) k χ I (s) |ψ(s, t)|2 ds dt,
q[ψ]− E 1 ψ ≧
2
R×ω 1 + (s − s0 )
where ch is the Hardy constant of Theorem 3. This proves that the threshold of the
spectrum of Q (and therefore of −∆Ω
D ) is greater than or equal to E 1 for sufficiently
small k.
In order to show that the whole interval [E 1 , ∞) belongs to the spectrum, it
is enough to construct an appropriate Weyl sequence in the infinite straight ends
of Ω. This concludes the proof of Theorem 1.
The proof of Theorem 2 is exactly the same, provided one chooses σ := θ̇ and
ij
k := κ1 ∞ +κ̇1 ∞ +κ2 ∞ . Indeed, all the above estimates are valid with (G 0 )
being now the matrix (30) after letting both κ1 and κ2 equal to zero, and with C1
and C3 being replaced by
2
C1 := 6 a 1 + a κ2 ∞ + a θ̇ ∞ ,
C3 := max 2, 1 + 2 a 2 θ̇ 2∞ ,
respectively. Here C1 can be further estimated by a constant independent of κ2
provided one restricts, for example, to κ2 ∞ < 1/a.
5. Discussion
We have established Hardy-type inequalities for twisted three-dimensional
tubes. As an application we have showed that the discrete eigenvalues of the Dirichlet Laplacian in mildly and locally bent tubes can be eliminated by an appropriate
twisting. However, we would like to point out that for σ = θ̇ , Theorems 3 and 4 can
be used to prove certain stability of transport in straight twisted tubes also against
other types of perturbations. For example against a local enlargement of the straight
tube, mentioned in Introduction, or in principle against any potential perturbation
which decays at least as O(s −2 ) at infinity, where s is the longitudinal coordinate
of the straight tube. The required decay at infinity is related to the decay of the
A Hardy Inequality in Twisted Waveguides
261
Hardy weight in Theorem 3, which is an inverse quadratic and cannot be improved
for locally twisted tubes.
For straight twisted tubes, the Hardy weight in the local inequality (21) of
Theorem 4 is given in terms of the function θ̇ and the constant λ. Roughly speaking,
the first tells us how fast the cross-section rotates, while the latter “measures” how
much the cross-section differs from a disc. The actual value of λ depends of course
on the geometry of ω.
The example of bent twisted tubes is of particular interest, since it shows the
important role of the torsion. Namely, Theorem 1 tells us that, whenever θ̇ = κ2 ,
the discrete eigenvalues in mildly curved tubes can be eliminated by torsion only.
Note that Theorem 1 also provides a better lower bound to the spectrum in mildly
bent tubes than that derived in [8].
Theorems 1 and 2 were proved for tubes about curves determined by (2). This
restriction was made in order to construct the tube uniquely from given curvature
functions by means of a uniquely determined Frenet frame. However, Theorems 1
and 2 will also hold for more general classes of tubes, namely, for those constructed
about curves possessing the distinguished Frenet frame and with curvatures decaying as O(s −2 ) at infinity, where s is the arc-length parameter of the curve.
At least from the mathematical point of view, it would be interesting to extend
Theorem 1 to higher dimensions. Here the main difficulty is that σ in the form
analogous to (17) will be in general a tensor depending also on the transverse
variables t. Nevertheless, a higher dimensional analogue of Theorem 2 is easy to
derive along the same lines as in the present paper, provided one restricts to rotations
of the cross-section just in one hyperplane.
Summing up, the twisting represents a repulsive geometric perturbation in the
sense that it eliminates the discrete eigenvalues in mildly curved waveguides.
Regarding the transport itself, an interesting open question is whether this also
happens to the singular spectrum possibly contained in the essential spectrum. It
would also be of a considerable interest to see what effect the twisting has on possible resonances, which might be induced by bending or by potential perturbations
of the waveguide.
As mentioned in the Introduction, the original motivation for our problem was
quantum-mechanical, for the one-particle Hamiltonian in a tube with Dirichlet
boundary conditions is a reasonable model for quantum waveguides [6,16]. It is
challenging to demonstrate the repulsive effect of twisting in other areas of physics
too, specifically in electromagnetic waveguides, acoustic waveguides or in waterflow pipes. While the effect of twisting could be easier to observe in experiments
with classical systems, theoretically the opposite is true, and the more complicated equations of motion and/or boundary conditions lead to completely different
mathematical problems.
Appendix A. Injectivity of the tube mapping
Let us conclude the paper by finding geometric conditions which guarantee the
basic hypotheses (11).
T. Ekholm, H. Kovařík & D. Krejčiřík
262
The first condition of (11) ensures that the mapping L is an immersion due
to (29). The second, injectivity condition requires to impose some global hypotheses
about the geometry of the curve. Our approach is based on the following lemma:
Lemma 4. Let Γ be determined by the curvature functions (2). Then for every
i ∈ {1, 2, 3} and all s1 , s2 ∈ R,
|ei (s2 ) − ei (s1 )| ≦ 2 ki min {|s2 − s1 |, |I |} ,
where


if i = 1,
κ1 ∞
ki := κ1 ∞ + κ2 ∞ if i = 2,


κ2 ∞
if i = 3.
Proof. It follows from the Serret–Frenet equations (3) and (2) that
s
s
2 2
|ei (s2 ) − ei (s1 )| ≦ 2 χ I ,
|ėi | ≦ 2 ki s1
s1
which immediately establishes the assertion.
⊓
⊔
As a consequence of Lemma 4, we get the inequality
ei (s2 ) · ei (s1 ) ≧ 1 − 2 |I |2 ki2 ,
i ∈ {1, 2, 3}.
(A.36)
In particular, since e1 is the tangent vector of Γ , we obtain that the curve is not
self-intersecting provided |I | κ1 ∞ < 1. A stronger sufficient condition ensures
the injectivity of L:
Proposition 1. Let Γ be determined by the curvature functions (2). Then the
hypotheses (11) hold true provided
max 4 |I |2 κ1 2∞ , 4 a (κ1 ∞ + κ2 ∞ ) < 1.
Proof. The idea is to observe that it is enough to show that the mapping Γt from R
to R3 defined by
Γt (s) := Γ (s) + tµ Rµν (s) eν (s)
is injective for any fixed t ∈ R2 such that |t| < a and arbitrary matrix-valued
function (Rµν ) : R → SO(2). Let us assume that there exist s1 < s2 such that
Γt (s1 ) = Γt (s2 ). Then
0 = Γ (s2 ) − Γ (s1 ) + tµ Rµν (s2 ) − Rµν (s1 ) eν (s1 )
+ Rµν (s2 ) [eν (s2 ) − eν (s1 )] .
Taking the inner product of both sides of the vector identity with the tangent
vector e1 (s1 ) and writing the difference Γ (s2 ) − Γ (s1 ) as an integral, we arrive at
the following scalar identity
s2
0=
e1 (s1 ) · e1 (ξ ) dξ + tµ Rµν (s2 ) [eν (s2 ) − eν (s1 )] · e1 (s1 ).
s1
A Hardy Inequality in Twisted Waveguides
263
Applying Lemma 4 together with the first inequality of (A.36), recalling the
orthogonality of (Rµν ) and using obvious estimates, we obtain
0 ≧ (s2 − s1 ) 1 − 2 |I |2 k12 − 2 a k2 .
This provides a contradiction for all curves satisfying the inequality of Proposition,
unless s1 = s2 . ⊓
⊔
Remark 2. The ideas of this Appendix are not restricted to the special class of
tubes about curves determined by (2). Indeed, assuming only the existence of an
appropriate Frenet frame for the reference curve (cf [5]), more general sufficient
conditions, involving integrals of curvatures, could be derived.
Acknowledgments. The authors are grateful to Timo Weidl for pointing out the presented
problem to them. The work has partially been supported by the Czech Academy of Sciences
and its grant agency within the projects IRP AV0Z10480505 and A100480501, by the project
LC06002 of the Ministry of Education, Youth and Sports of the Czech Republic, and by
DAAD within the project D-CZ 5/05-06. T.E. has partially been supported by the ESF European programme SPECT and D.K. has partially been supported by FCT/POCTI/FEDER,
Portugal.
References
1. Borisov, D., Ekholm, T., Kovařík, H.: Spectrum of the magnetic Schrödinger operator
in a waveguide with combined boundary conditions. Ann. H. Poincaré 6, 327–342
(2005)
2. Borisov, D., Exner, P., Gadyl’shin, R.R., Krejčiřík, D.: Bound states in weakly
deformed strips and layers. Ann. Henri Poincaré 2, 553–572 (2001)
3. Bouchitté, G., Mascarenhas, M.L., Trabucho, L.: On the curvarture and torsion
effects in one dimensional waveguides. Control, Optim. Calc. Var. 13(4), 793–808
(2007)
4. Bulla, W., Gesztesy, F., Renger, W., Simon, B.: Weakly coupled bound states in
quantum waveguides. Proc. Am. Math. Soc. 125(5), 1487–1495 (1997)
5. Chenaud, B., Duclos, P., Freitas, P., Krejčiřík, D.: Geometrically induced discrete
spectrum in curved tubes. Differ. Geom. Appl. 23(2), 95–105 (2005)
6. Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two
and three dimensions. Rev. Math. Phys. 7, 73–102 (1995)
7. Ekholm, T., Kovařík, H.: Stability of the magnetic Schrödinger operator in a
waveguide. Comm. Partial Differ. Equ. 30, 539–565 (2005)
8. Exner, P., Freitas, P., Krejčiřík, D.: A lower bound to the spectral threshold in curved
tubes. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460(2052), 3457–3467 (2004)
9. Exner, P., Šeba, P.: Bound states in curved quantum waveguides. J. Math. Phys. 30,
2574–2580 (1989)
10. Goldstone, J., Jaffe, R.L.: Bound states in twisting tubes. Phys. Rev. B. 45, 14100–
14107 (1992)
11. Grushin, V.V.: On the eigenvalues of finitely perturbed Laplace operators in infinite
cylindrical domains. Math. Notes 75(3), 331–340 (2004). Grushin, V.V.: Translation
from Mat. Zametki 75(3), 360–371 (2004)
12. Grushin, V.V.: Asymptotic behavior of the eigenvalues of the Schrödinger operator
with transversal potential in a weakly curved infinite cylinder. Math. Notes 77(5), 606–
613 (2005). Grushin, V.V.: Translation from Mat. Zametki 77(5), 656–664 (2005)
264
T. Ekholm, H. Kovařík & D. Krejčiřík
13. Hurt, N.E.: Mathematical Physics of Quantum Wires and Devices. Kluwer, Dordrecht,
2000
14. Klingenberg, W.: A course in differential geometry. Springer, New York, 1978
15. Krejčiřík, D., Kříž, J.: On the spectrum of curved quantum waveguides. Publ. RIMS,
Kyoto University. 41(3), 757–791 (2005)
16. Londergan, J.T., Carini, J.P., Murdock, D.P.: Binding and Scattering in TwoDimensional Systems. LNP, vol. m60, Springer, Berlin, 1999
17. Renger, W., Bulla, W.: Existence of bound states in quantum waveguides under weak
conditions, Lett. Math. Phys. 35, 1–12 (1995)
Lund University,
Centre for Mathematical Sciences,
Box 118,
221 00 Lund, Sweden.
e-mail: [email protected]
and
Institute of Analysis, Dynamics and Modeling,
Faculty of Mathematics and Physics,
Stuttgart University, PF 80 11 40,
70569 Stuttgart, Germany.
e-mail: [email protected]
and
Department of Theoretical Physics,
Nuclear Physics Institute, Academy of Sciences,
250 68 Řež near Prague, Czech Republic.
e-mail: [email protected]
(Received June 10, 2005 / Accepted February 1, 2007)
Published online February 5, 2008 – © Springer-Verlag (2008)
2.4
Twisted Dirichlet-Neumann waveguide
To appear in Math. Nachr. Preprint: arXiv: math-ph/0603076.
119
A Hardy inequality in a twisted
Dirichlet-Neumann waveguide
H. Kovařı́k and D. Krejčiřı́k
23 February 2006
Dedicated to Pavel Exner on the occasion of his 60th birthday.
Abstract
We consider the Laplacian in a straight strip, subject to a combination
of Dirichlet and Neumann boundary conditions. We show that a switch
of the respective boundary conditions leads to a Hardy inequality for the
Laplacian. As a byproduct of our method, we obtain a simple proof of a
theorem of Dittrich and Křı́ž [5].
1
Introduction
The connection between spectral properties of the Laplace operator in a waveguide-type domain, the domain geometry and various boundary conditions has
been intensively studied in the last years, cf [6, 14, 12] and references therein.
Particular attention has been paid to the geometrically induced discrete spectrum of the Dirichlet Laplacian in curved tubes of uniform cross-section [9, 10,
16, 6, 4] or in straight tubes with a local deformation of the boundary [3, 2].
Roughly speaking, it has been shown that a suitable bending or a local enlargement of a straight waveguide represents an effectively attractive perturbation
and leads thus to the presence of eigenvalues below the essential spectrum of
the Laplacian.
On the other hand, recently it has been observed in [8] that a local rotation
of a non-circular cross-section of a three-dimensional straight tube creates a
kind of repulsive perturbation. Namely, this type of deformation, called twist,
gives rise to a Hardy inequality for the Dirichlet Laplacian. This avoids, up to
some extent, the existence of discrete spectrum in the presence of an additional
attractive perturbation, the bending or local enlargement being two examples.
We refer to [8] for more details and possible higher-dimensional extensions.
The purpose of the present note is to demonstrate an analogous effect of twist
in a two-dimensional waveguide with combined Dirichlet and Neumann boundary conditions. In this case the twist is represented by a switch of the boundary
1
conditions at a given point, cf Figure 1. More precisely, given a real number ε
and a positive number a, let −∆ε be the Laplacian in the strip R × (−a, a), subject to Dirichlet boundary conditions on (−∞, −ε) × {−a} ∪ (ε, +∞) × {a} and
Neumann boundary conditions on (−ε, +∞)×{−a}∪(−∞, ε)×{a}, cf Figure 2.
It can be seen by a simple Neumann bracketing that the spectrum of −∆ε coincides with the interval [π 2 /(4a)2 , +∞) for all non-positive ε. Our main result
shows that for ε equal to zero the operator −∆0 satisfies the following Hardy
type inequality in the sense of quadratic forms:
−∆0 −
π 2
≥ ρ(·) ,
4a
(1)
where ρ : R × (−a, a) → R is a positive function.
We would like to emphasize that in the situation where the boundary conditions are not exchanged – i.e. the Laplacian in R × (−a, a) with uniform
Dirichlet boundary conditions on one connected part of the boundary and Neumann boundary conditions on the other one, cf the upper waveguide in Figure 1
– the essential spectrum coincides with the essential spectrum of our waveguide,
but the inequality (1) fails to hold for any non-trivial ρ ≥ 0. The latter can
be shown by a simple test-function argument. In other words, the switch of
the boundary conditions creates a kind of repulsive perturbation represented by
the function ρ. This leads to a certain stability of the spectrum similar to the
one observed in [8]. In particular, it follows from (1) that perturbing −∆0 by a
sufficiently small attractive perturbation, the discrete spectrum remains empty.
One example of attractive perturbation is changing the boundary conditions
by increasing the parameter ε, cf Figure 2. Due to the switch of the boundary
conditions, the discrete eigenvalues do not appear for any positive ε, but only
when ε exceeds certain critical value εc > 0. This effect was already observed
by Dittrich and Křı́ž in [5]. Their result is obtained by a tedious decomposition
of the Laplacian into the “transverse basis” and this also provides an estimate
on the critical value εc for which the eigenvalues emerge from the essential
spectrum:
0.16 a < εc < 0.68 a .
(2)
Since the proof of our Hardy inequality (1) can be easily carried over to the
case when ε is positive and small enough, we get as a byproduct of our method
an alternative estimate on εc , too. The latter is worse than the one presented
in [5], but on the other hand much simpler to obtain.
Finally, let us mention that Hardy inequalities for Schrödinger operators in
two dimensions can be achieved by adding an appropriate local magnetic field
to the system, too. This was first observed in [13] and later modified in [7]
for Schrödinger operators in waveguides, cf also [1]. Curved waveguides in a
homogeneous magnetic field have been recently studied in [15].
2
2
Main results and ideas
The Laplacian −∆ε is defined as the unique self-adjoint operator associated
with the closure of the quadratic form Qε defined in L2 R × (−a, a) by
Z
(3)
|∂1 ψ(x, y)|2 + ∂2 ψ(x, y)|2 dx dy
Qε [ψ] :=
R×(−a,a)
and by the domain D(Qε ) which consists of restrictions to R×(−a, a) of infinitely
smooth functions with compact support in R2 and vanishing on the part of the
boundary where the Dirichlet boundary conditions are imposed (cf [5] for more
details). We are interested in the shifted quadratic form Q̃ε defined on the form
domain D(Qε ) by the prescription
π 2 Z
Q̃ε [ψ] := Q[ψ] −
|ψ(x, y)|2 dx dy .
(4)
4a
R×(−a,a)
If ε is negative, so that the opposite Dirichlet boundary conditions overlap, one can estimate the second term in (3) by the lowest eigenvalue of the
Laplacian in the cross-section of length 2a, subject to Dirichlet-Dirichlet or
Dirichlet-Neumann boundary conditions. Neglecting the first term in (3), this
immediately yields
π 2
π 2
−∆ε −
≥ 3
χ(ε,−ε)×(−a,a) (·)
if
ε<0
(5)
4a
4a
in the sense of quadratic forms. Here χM denotes the characteristic function
of a set M . The right hand side provides a non-negative Hardy weight in this
case.
Of course, the trivial estimate leading to (5) is not useful for non-negative ε,
in which case other methods have to be used. In this paper we get:
Theorem 1. Given a real number ε and a positive number a, let −∆ε be the
Laplacian in the strip R × (−a, a), subject to Dirichlet boundary conditions
on (−∞, −ε) × {−a} ∪ (ε, +∞) × {a} and Neumann boundary conditions on
(−ε, +∞) × {−a} ∪ (−∞, ε) × {a}.
(i) There exists a positive constant c such that the inequality
π 2
≥ c χω (·)
−∆0 −
4a
(6)
holds in the sense of quadratic forms. Here ω ⊇ (−a, a) × (−a, a) and
π 2
,
c ≥ s1
4a
where s1 is the smallest root of the equation
√
p
√
π 1−s
√
1 − s tanh
= 1/2 + s tan
2 2
3
π
!
p
1/2 + s
√
.
2 2
(7)
(ii) There exists a positive constant εc ≥ t1 a such that
σ(−∆ε ) = π 2 /(4a)2 , ∞
for all ε ≤ εc . Here t1 is the smallest positive root of the equation
p
π (1 − t)
π (1 + t)
√
√
= 1/2 tan
.
tanh
2 2
2 2
(8)
The first result, i.e. the Hardy inequality for −∆0 , is new. On the other
hand, a positive lower bound on εc has already been established in [5], cf (2).
In [5] the authors also find the numerical value εc ≈ 0.52 a. We have s1 ≈ 0.039
and t1 ≈ 0.061, and these numbers cannot be much improved by our method
(cf the end of Section 4 for more details).
Although the effect which causes (6) is very similar to the twist studied in [8],
the methods used in the respective proofs are completely different. The reason is
that in our case the twist represents a singular deformation in the sense that it is
discontinuous and occurs at one point only. Our main idea to prove Theorem 1
is to introduce rotated Cartesian coordinates in which one can easily employ
the repulsive interaction due to the proximity of opposite Dirichlet boundary
conditions, cf Figure 3. This is done in Section 3 where the initial problem is
reduced to an ordinary differential equation. The latter is then investigated in
Section 4 by standard methods for one-dimensional Schrödinger operators.
Note that Theorem 1 contains a weaker version of inequality (1), namely
with a compactly supported Hardy weight. However, (1) can be easily deduced
from it:
Corollary 1. Inequality (1) holds true with the function ρ given by
−1
ch
ρ(x, y) :=
,
ch := max 16, c−1 (2 + 16/a2 )
,
1 + x2
where c is the constant from Theorem 1.
A short proof of Corollary 1, based on the classical one-dimensional Hardy
inequality, is given in the concluding Section 5.
3
Reduction to a one-dimensional problem
Let (x, y) ∈ R × (−a, a). We introduce rotated Cartesian coordinates (u, v) by
the change of variables
(x, y) = f (u, v) := u cos θ + v sin θ, −u sin θ + v cos θ ,
(9)
where θ ∈ (0, π/2). Clearly, the mapping f : Ω → R × (−a, a) is a diffeomorphism with the preimage
Ω := f −1 (R × I) = (u, v) ∈ R2 | u− (v) < u < u+ (v)
= (u, v) ∈ R2 | v− (u) < v < v+ (u) ,
4
where
±a + u sin θ
±a + v cos θ
,
v± (u) :=
.
sin θ
cos θ
Introducing the (unitary) change of trial function ψ 7→ ψ ◦ f := φ into the
functional (3), we find
Z
(10)
|∂1 φ(u, v)|2 + ∂2 φ(u, v)|2 du dv .
Qε [φ ◦ f −1 ] =
u± (v) :=
Ω
From the formulae
u ± a sin θ
φ u, v± (u) = ψ
, ±a ,
cos θ
φ u± (v), v = ψ
v ± a cos θ
, ∓a ,
sin θ
we observe the two following properties, respectively. First, v 7→ φ(u, v) with u
fixed satisfies Dirichlet boundary conditions at both boundary points v± (u) if,
and only if,
|u| < u0 := a sin θ − ε cos θ ;
(11)
otherwise it satisfies a combination of Dirichlet and (generalized) Neumann
boundary conditions. Second, u 7→ φ(u, v) with v fixed satisfies a combination
of Dirichlet and (generalized) Neumann boundary conditions, if, and only if,
|v| > v0 := a cos θ + ε sin θ ;
(12)
otherwise it satisfies (generalized) Neumann boundary conditions (i.e. none).
While v0 is positive by definition, we need to assume that
ε < a tan θ
(13)
in order to ensure the positivity of u0 .
We proceed by estimating the form (10) as follows. We estimate the second
term in (10) by the lowest eigenvalue of the Laplacian in the cross-section of
length v+ (u) − v− (u) = 2a/ cos θ, subject to the boundary conditions of the
type that v 7→ φ(u, v) satisfies. We also estimate the first term in (10) by
the lowest eigenvalue of the Laplacian in the cross-section of length u+ (v) −
u− (v) = 2a/ sin θ, subject to the boundary conditions of the type that u 7→
φ(u, v) satisfies, but only in the subset of Ω where |u| > u0 and |v| > v0 . That
is,
Z
Z
Z
|φ|2 ,
(14)
|φ|2 − q−
|∂1 φ|2 + q+
Q̃ε [φ ◦ f −1 ] ≥
Ω1
Ω1 ∪Ω2
Ω2
where
Ω1 := {(u, v) ∈ Ω | |u| < u0 } ,
and
q+ :=
Ω2 := {(u, v) ∈ Ω | |v| < v0 , |u| > u0 } ,
π 2
(4 cos2 θ − 1) ,
4a
5
q− :=
π 2
sin2 θ .
4a
(15)
Hereafter we further restrict the angle θ by the requirement
θ ∈ (0, π/3) ,
(16)
so that the term q+ is positive.
We use the intermediate bound (14) as the starting point of the reduction
to a one-dimensional problem. Let us introduce the disjoint sets
Ω′1 := {(u, v) ∈ Ω | |u| < u0 , |v| > v0 } ,
Ω′2 := {(u, v) ∈ Ω | |v| < v0 } ,
and note that the inclusions Ω′1 ⊂ Ω1 and Ω′2 ⊂ Ω1 ∪ Ω2 hold. Consequently,
under the assumption (16), (14) implies the cruder bound
Z
Z
−1
2
λ(v) |φ(u, v)|2 du dv ,
(17)
Q̃ε [φ ◦ f ] ≥ q+
|φ(u, v)| du dv +
Ω′2
Ω′1
where λ(v) is the lowest eigenvalue of the one-dimensional Neumann Schrödinger
operator with the step-like potential
V (u, v) := q+ χ(−u0 ,u0 ) (u) − q− χ(u− (v),−u0 )∪(u0 ,u+ (v)) (u) ,
More precisely,
i
R u+ (v) h ′
2
2
du
|ϕ
(u)|
+
V
(u,
v)
|ϕ(u)|
u− (v)
λ(v) := inf
,
R
u+ (v)
ϕ
|ϕ(u)|2 du
u− (v)
(18)
where the infimumis taken over all non-zero functions from the Sobolev space
W 1,2 u− (v), u+ (v) .
The formula (17) together with (18) transfers the initial two-dimensional
problem into the study of an ordinary differential equation. That is, it remains
to investigate the function v 7→ λ(v).
4
Study of the one-dimensional problem
First of all, we observe that v 7→ λ(v) is an even function with values in the
open interval (q− , q+ ) due to (16). Furthermore, its minimum is attained at the
boundary points v = ±v0 :
Lemma 1. One has
inf
v∈(−v0 ,v0 )
λ(v) = λ(v0 ) .
Proof. Let h, l and δ be positive numbers such that δ < l. For any real c, we
consider the one-dimensional Schrödinger operator
Hc := −∆ + h χ(c,c+δl)
in
L2 (0, l) ,
subject to Neumann boundary conditions. (Hc is introduced in a standard way
through the associated quadratic form defined in W 1,2 ((0, l)).) Let us show
∀c ∈ (0, l − δl),
inf σ(Hc ) ≥ inf σ(H0 ) ,
6
(19)
which is equivalent to the statement of the Lemma.
The reader is advised to consult Figure 4 for the following construction.
Given c ∈ (0, l − δl), we find α1 , α2 ∈ (0, 1) such that
α1 + α2 = 1
c
α1
=
.
α2
l − (c + δl)
and
We also define parameters δ1 , δ2 ∈ (0, δ) by the equations
δ1 + δ2 = δ
and
δ1
α1
=
.
δ2
α2
It follows that α1 l = c + δ1 l. Let t∗ := α1 l ∈ (0, l).
The minimax principle yields
inf σ(Hc ) ≥ inf σ(HcN ) ,
where HcN is the operator obtained from Hc by imposing an additional Neumann
boundary condition at the point t∗ . HcN is a direct sum of two operators, which
are unitarily equivalent to
T1 := −∆ + h χ(0,δ1 l)
in
L2 (0, α1 l) ,
T2 := −∆ + h χ(0,δ2 l)
in
L2 (0, α2 l) ,
respectively, both subject to Neumann boundary conditions. Hence,
σ(HcN ) = σ(T1 ) ∪ σ(T2 ) .
(20)
Obvious changes of variable show that that T1 and T2 are unitarily equivalent
to the operators
T̂1 := −(δ/δ1 )2 ∆ + h χ(0,δl)
in
L2 (0, l) ,
T̂2 := −(δ/δ2 )2 ∆ + h χ(0,δl)
in
L2 (0, l) ,
respectively, both subject to Neumann boundary conditions. Consequently,
T̂1 ≥ H0
and
T̂2 ≥ H0
in the sense of quadratic forms. This together with (20) implies (19).
As a consequence of (17) and the above Lemma, we therefore obtain
Z
|φ(u, v)|2 du dv .
Q̃ε [φ ◦ f −1 ] ≥ λ(v0 )
Ω′
We now turn to a more quantitative study of λ(v0 ). The eigenvalue problem
associated with (18) can be solved explicitly in the intervals where the potential V is constant. Matching these solutions in the discontinuity points of V ,
7
one easily finds that λ(v0 ) coincides with the smallest root λ ∈ (q− , q+ ) of the
equation
g1 (λ, ε, θ) = g2 (λ, ε, θ) ,
(21)
where
p
p
q+ − λ tanh 2u0 q+ − λ ,
p
p
g2 (λ, ε, θ) := q− + λ tan 2v0 cot θ q− + λ .
g1 (λ, ε, θ) :=
Recall that q+ , q− and u0 , v0 are introduced in (15) and (11)–(12), respectively.
Of course, g2 is not defined for all the values
λ, ε, θ, and we
pof the parameters
should rather multiply (21) by cos 2v0 cot θ q− + λ , but the resulting (regular) equation cannot be satisfied if the cosine equals zero, so we can leave (21)
in the present form.
Let us first consider the case ε = 0. A necessary condition to guarantee the
eligibility of our method to prove Theorem 1 is that λ(v0 ) is positive for certain
angle θ satisfying (16). A numerical study of (21) shows that λ(v0 ) achieves
its maximum, given approximately by 0.040 π 2/(4a)2 , for the angle θ ≈ 0.774.
Observing that the optimal angle is close to π/4 ≈ 0.785, let us fix henceforth:
θ = π/4 .
(22)
Since λ 7→ g1 (λ, 0, π/4) is decreasing and continuous, λ 7→ g2 (λ, 0, π/4) is increasing and continuous, and at λ = 0 we have
√
g1 (0, 0, π/4) √
(23)
= 2 tanh 2 π/4 > 1 ,
g2 (0, 0, π/4)
it follows that λ(v0 ) is indeed positive for the choice (22). As for the numerical
value, it is straightforward to check that (21) reduces to (7) and we find that
the smallest root s1 of the latter equals approximately 0.039. At the same time,
q+ = π 2 /(4a)2 for the choice (22). Summing up, (17) implies
π 2 Z
Q̃ε [φ ◦ f −1 ] ≥ s1
|φ(u, v)|2 du dv ,
′
′
4a
Ω1 ∪Ω2
provided the angle θ is chosen according to (22). It remains to realize that
f Ω′1 ∪ Ω′2 ⊃ (−a, a) × (−a, a) ,
where f is given by (9), in order to establish (i) of Theorem 1.
In the case of positive ε, we put λ equal to zero in (21) and look for the smallest positive ε satisfying the equation (21). This root satisfies the restriction (13)
because ε 7→ g1 (0, ε, π/4) is decreasing and continuous, ε 7→ g2 (0, ε, π/4) is increasing and continuous, g1 (0, a, π/4) = 0, g2 (0, ε, π/4) tends to +∞ as ε → a,
and we have (23) for ε = 0. It is straightforward to check that (21) reduces
to (8) for the choice (22) and the smallest positive root t1 of the latter equals
approximately 0.061. Again, a more detailed numerical study of (21) shows
that the best result reachable by the present method gives εc ≈ 0.063 a with the
optimal angle θ ≈ 0.759.
This concludes the proof of Theorem 1.
8
5
Proof of Corollary 1
The local Hardy inequality (6) is equivalent to
Z
Z
Z
|∂1 ψ|2 +
|ψ|2 ≤
(−a,a)×(−a,a)
R×(−a,a)
R×(−a,a)
|∂2 ψ|2 −
π 2 Z
|ψ|2
4a
R×(−a,a)
for any ψ ∈ D(Qε ) ⊂ W 1,2 R × (−a, a) . Here the sum of the last two terms
on the right hand side is non-negative due to the boundary conditions that ψ
satisfies. Consequently, Corollary 1 follows at once by means of the following
Hardy-type inequality for a Schrödinger operator in a strip with the potential
being a characteristic function:
Lemma 2. For any ψ ∈ W 1,2 R × (−a, a) ,
Z
Z
Z
|∂1 ψ|2 + 2 + 64/|J|2
w−2 |ψ|2 ≤ 16
|ψ|2 ,
R×(−a,a)
J×(−a,a)
R×(−a,a)
p
where w(x, y) := 1 + (x − x0 )2 , J is any bounded subinterval of R and x0 is
the mid-point of J.
This Lemma can be established
of the classical oneR
R quite easily by means
dimensional Hardy inequality R x−2 |v(x)|2 dx ≤ 4 R |v ′ (x)|2 dx valid for any
v ∈ W 1,2 (R) with v(0) = 0 and Fubini’s theorem; we refer the reader to [8,
Sec. 3.3] or [11, proof of Lem. 2] for more details.
Acknowledgement
The work has partially been supported by the Czech Academy of Sciences and
its Grant Agency within the projects IRP AV0Z10480505 and A100480501, and
by DAAD within the project D-CZ 5/05-06.
References
[1] D. Borisov, T. Ekholm, and H. Kovařı́k, Spectrum of the magnetic
Schrödinger operator in a waveguide with combined boundary conditions,
Ann. H. Poincaré 6 (2005), 327–342.
[2] D. Borisov, P. Exner, R. Gadyl’shin, and D. Krejčiřı́k, Bound states in
weakly deformed strips and layers, Ann. H. Poincaré 2 (2002), 553–572.
[3] W. Bulla, F. Gesztesy, W. Renger, and B. Simon, Weakly coupled bound
states in quantum waveguides, Proc. Amer. Math. Soc. 125 (1997), 1487–
1495.
[4] B. Chenaud, P. Duclos, P. Freitas, and D. Krejčiřı́k, Geometrically induced
discrete spectrum in curved tubes, Differential Geom. Appl. 23 (2005), no. 2,
95–105.
9
[5] J. Dittrich and J. Křı́ž, Bound states in straight quantum waveguides with
combined boundary condition, J. Math. Phys. 43 (2002), 3892–3915.
[6] P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73–102.
[7] T. Ekholm and H. Kovařı́k, Stability of the magnetic Schrödinger operator
in a waveguide, Comm. in PDE 30 (2005), 539–565.
[8] T. Ekholm, H. Kovařı́k, and D. Krejčiřı́k, A Hardy inequality in twisted
waveguides, submitted; preprint on [math-ph/0512050] (2005).
[9] P. Exner and P. Šeba, Bound states in curved quantum waveguides,
J. Math. Phys. 30 (1989), 2574–2580.
[10] J. Goldstone and R. L. Jaffe, Bound states in twisting tubes, Phys. Rev. B
45 (1992), 14100–14107.
[11] D. Krejčiřı́k, Hardy inequalities for strips on ruled surfaces, J. Inequal.
Appl., to appear; preprint on [math.SP/0511257] (2005).
[12] D. Krejčiřı́k and J. Křı́ž, On the spectrum of curved quantum waveguides,
Publ. RIMS, Kyoto University 41 (2005), no. 3, 757–791.
[13] A. Laptev and T. Weidl, Hardy inequalities for magnetic Dirichlet forms,
Oper. Theory Adv. Appl. 108 (1999), 299–305.
[14] J. T. Londergan, J. P. Carini, and D. P. Murdock, Binding and scattering
in two-dimensional systems, LNP, vol. m60, Springer, Berlin, 1999.
[15] O. Olendski and L. Mikhailovska, Curved quantum waveguides in uniform
magnetic fields, Phys. Rev. B 72 (2005), 235314.
[16] W. Renger and W. Bulla, Existence of bound states in quantum waveguides
under weak conditions, Lett. Math. Phys. 35 (1995), 1–12.
List of Figures
1
2
3
4
We consider the lower waveguide as a twist perturbation of the
upper one, the twist being defined as a switch of Dirichlet (thick
lines) to Neumann (thin lines) boundary conditions, and vice
versa, at one point. . . . . . . . . . . . . . . . . . . . . . . . . .
The geometry of our waveguide. The Dirichlet and Neumann
boundary conditions are denoted by thick and thin lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rotating the Cartesian coordinate system by an appropriate angle θ, one can easily employ the repulsive interaction due to the
proximity of opposite Dirichlet boundary conditions (thick lines).
The construction used in the proof of Lemma 1. . . . . . . . . .
10
11
11
12
12
Figure 1: We consider the lower waveguide as a twist perturbation of the upper
one, the twist being defined as a switch of Dirichlet (thick lines) to Neumann
(thin lines) boundary conditions, and vice versa, at one point.
y
a
-e
e
x
-a
Figure 2: The geometry of our waveguide. The Dirichlet and Neumann boundary conditions are denoted by thick and thin lines, respectively.
11
v
v0
q
u
u0
-u0
-v0
Figure 3: Rotating the Cartesian coordinate system by an appropriate angle θ,
one can easily employ the repulsive interaction due to the proximity of opposite
Dirichlet boundary conditions (thick lines).
d2 l
d1 l
h
0
c
t*
c+dl
a1 l
l
a2 l
Figure 4: The construction used in the proof of Lemma 1.
12
t
2.5
Periodically twisted tube
Published in Lett. Math. Phys. 73 (2005) 183–192.
131
Letters in Mathematical Physics (2005) 73:183–192
DOI 10.1007/s11005-005-0016-8
© Springer 2005
Spectrum of the Schrödinger Operator
in a Perturbed Periodically Twisted Tube
P. EXNER1,2 and H. KOVAŘÍK1,3
1
Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences,
CZ-25068 Řež near Prague, Czech Republic. e-mail: [email protected]
2
Doppler Institute, Czech Technical University, Břehová 7, CZ-11519 Prague,
Czech Republic
3
Institute for Analysis, Dynamics and Modeling, Faculty of Mathematics and Physics,
Stuttgart University, PF 80 11 40, D-70569 Stuttgart, Germany.
e-mail: [email protected]
(Received: 6 May 2005; revised version: 22 August 2005)
Abstract. We study Dirichlet Laplacian in a screw-shaped region, i.e. a straight twisted
tube of a non-circular cross section. It is shown that a local perturbation which consists
of “slowing down” the twisting in the mean gives rise to a non-empty discrete spectrum.
Mathematics Subject Classification: 35P05, 81Q10.
Key words. Schrödinger operator, discrete spectrum, twisting.
1. Introduction
Existence of geometrically-induced bound states in infinitely extended regions of
tubular shape was noticed at the end of the 1980s, first in the two-dimensional situation [9], and studied intensively since then – see [5, 11, 13, 15], and more recently
[4, 8]. The effect attracted attention not only due to its nonclassical nature but
also because it concerned a natural model of semiconductor structures which were
likely to be construction blocks of the next-generation microelectronics. This fact
alone imparts a strong motivation to study different mechanisms which can lead
to occurrence of such “trapped modes” or, on the contrary, to destroy them.
Most efforts have so far been devoted to the situation wherein the effective
attractive interaction, which led to the existence of a discrete spectrum, came from
tube bends. This is not the only possibility, however. In the practically important
case where the tube is embedded in Rd with d3, one can ask about the effect
of its twisting on the spectrum. In the works quoted above it was not important
since the tubes considered were either circular or twisted and aligned in a particular way with the tube axis torsion; in that case the twist did not affect the result
in the leading order.
184
P. EXNER AND H. KOVAŘÍK
Tubes of a non-circular cross section and related systems were considered for the
first time in [2, 3] where the authors noted that the twisting might act effectively
as a repulsive interaction. A more profound analysis of this effect was undertaken
only recently. Inspired by the existence of a magnetic Hardy-type inequality in
waveguides [6], the authors of [7] analyzed the generic case of a local tube twist
and found that it indeed gave rise to a repulsive effective interaction which can
destroy weakly bound states coming from other perturbations. In other words, the
result of [7] shows that a local twist, which does not affect the essential spectrum,
stabilizes the transport in the tube with non-circular cross-section. This is interesting, since twisting is a geometrical perturbation which does not change the shape
of the cross-section; in particular, it does not change the local volume of the tube.
The general motivation stated above suggests the opposite question, namely
whether twisting can be used to create bound states. The way to achieve this is
to modify the essential spectrum by taking a twist which extends along the whole
tube. Our main aim in this letter is to show that if we perturb a screw-shaped tube
locally in such a way that the repulsion is weakened, e.g. by a local slowdown of
the twist, one may expect a binding effect.
We start with a periodically twisted tube and find the new threshold of the
essential spectrum. Then we will show, under mild regularity assumptions, that
for the existence of the discrete spectrum it is sufficient if the twisting is slowed
down locally in the mean. Moreover, a natural analogy with the one-dimensional
Schrödinger operator theory suggests that the effect might survive in the critical
case when the mean value of the twist variation is zero. It is not a reliable guide,
of course, because sometimes in similar situations critical bound states are absent
[1]; nevertheless, here we are able to demonstrate that discrete spectrum for a critical twist perturbation is still non-empty. Let us review the contents of the latter.
We will introduce the needed notation in the next section, then analyze the spectrum in the periodic case. Our main results are given in Section 4, specifically in
Theorems 2 and 3 for the non-critical and critical situations, respectively.
2. Preliminaries
First we fix the notation. Let ω be an open bounded and connected set in R2 and
let θ be a differentiable function from R to R. For s ∈ R and t := (t2 , t3 ) ∈ ω we
define the mapping L from R × ω to R3 by
L(s, t) = (s, t2 cos θ (s) + t3 sin θ (s), t3 cos θ (s) − t2 sin θ (s)) .
(1)
The image L(R × ω) is a tube in R3 which is twisted unless the function θ is constant. A case of particular interest is a screw-shaped tube corresponding to a linear
θ . We fix a positive constant β0 and define the tube 0 by
0 := L0 (R × ω) ,
SPECTRUM OF THE SCHRÖDINGER OPERATOR
185
where
L0 (s, t) := (s, t2 cos(β0 s) + t3 sin(β0 s), t3 cos(β0 s) − t2 sin(β0 s));
it will play the role of the unperturbed system. The operator we will be concerned
with is the Dirichlet Laplacian H0 on L2 (0 ), i.e. the self-adjoint operator associated with the closed quadratic form
(2)
|∇ψ|2 ds dt , ∀ ψ ∈ D(Q0 ) = H01 (0 ) .
Q0 [ψ] :=
0
3. Spectrum of H0
Given ψ ∈ C0∞ (R × ω), it is useful to introduce the following shorthand,
ψτ′ := t2 ∂t3 ψ − t3 ∂t2 ψ .
(3)
A simple substitution of variables shows that
Q0 [ψ] =
|∇t ψ|2 + |∂s ψ + β0 ψτ′ |2 ds dt ,
R×ω
where
∇t ψ := (∂t2 ψ, ∂t3 ψ).
In other words, the operator H0 acts on its domain in L2 (0 ) as
H0 = −∂2t2 − ∂2t3 + (−i∂s − i β0 (t2 ∂t3 − t3 ∂t2 ))2
Since β0 is independent of s we are able to employ a partial Fourier transformation Fs given by
1
e−i ps ψ(s, t)ds,
(Fs ψ)(p, t) = ψ̂(p, t) = √
2π R
which allows us to rewrite the quadratic form as
|∇t ψ̂|2 + |i p ψ̂ + β0 ψ̂τ′ |2 dp dt
Q0 [ψ̂] =
R×ω
for a suitably regular ψ. Since the transformation Fs extends to a unitary operator
on L2 (R × ω), the operator H0 is unitarily equivalent to the direct integral
⊕
h(p) dp
(4)
R
with the fibre operator
h(p) = −∂2t2 − ∂2t3 + (p − i β0 (t2 ∂t3 − t3 ∂t2 ))2
(5)
186
P. EXNER AND H. KOVAŘÍK
on L2 (ω) subject to Dirichlet boundary conditions at ∂ω. Introducing the polar
coordinates (r, α) on ω, we can rewrite h(p) as follows
h(p) = −ωD + (p − i β0 ∂α )2 ,
(6)
where −ωD denotes the Dirichlet Laplacian in r and α. Since h(p) is a sum of
−ωD and a positive perturbation, it follows easily from the minimax principle that
its spectrum is purely discrete. Let us denote the eigenvalues of h(p) by εn (p) and
the respective eigenfunctions by ψn (p), i.e.
h(p) ψn (p) = εn (p) ψn (p) .
LEMMA 1. Every εn (·), n ∈ N, is a real-analytic function of p and
lim εn (p) → ∞ .
(7)
p→±∞
Proof. It is not difficult to check that the quadratic form associated with the
operator h(0) defined on the form domain H01 (0 ) is non-negative and closed.
This implies that h(0) is self-adjoint on its natural domain which we denote as
D(0). Let us formally expand the square in Equation (6) and write h(p) as
h(p) = h(0) + p2 − 2i p β0 ∂α .
Denote the resolvent of h(0) at a point z ∈ C by Rz , i.e. Rz = (h(0) − z)−1 . Then we
have for any ϕ ∈ C0∞ (ω) the following estimate
∂α ϕ2 (ϕ, h(0)ϕ) = (Rz (h(0) − z)ϕ, h(0)ϕ)
Rz h(0)ϕ2 + |z| (ϕ, Rz̄ h(0)ϕ)
C(z) h(0)ϕ2 + |z|2 C(z) ϕ2 ,
where C(z) → 0 as ℑz → ∞. Consequently, i ∂α is h(0)-bounded with the relative
bound zero which implies that the domain of h(p) coincides with D(0) and the
vector h(p)φ is analytic as a function of p for every φ ∈ D(0) (since p 2 is clearly
analytic). From [12], p. 375 and 385, it thus follows that {h(p) : p ∈ R} is a selfadjoint analytic family of type A and that all the εn (·) are real-analytic functions
of p.
To prove the second statement of the lemma, let us first define the cross-section
radius with respect to the rotation axis,
a := sup |t| .
t∈ω
We observe that for any ϕ ∈ C0∞ (ω) we have a trivial pointwise inequality,
|2p β0 ϕ̄ ∂α ϕ| p 2
β02
β02 + a −2
|ϕ|2 + (β02 + a −2 ) |∂α ϕ|2 ,
187
SPECTRUM OF THE SCHRÖDINGER OPERATOR
which implies that
1
2
2
2
(ϕ, h(p) ϕ) =
|∂r ϕ| + 2 |∂α ϕ| + |(p − i β0 ∂α )ϕ| r dr dα
r
ω 2
|∂r ϕ| + a −2 |∂α ϕ|2 + p 2 |ϕ|2 − |2pβ0 ϕ̄∂α ϕ| + β02 |∂α ϕ|2 rdr dα
ω
1
2
p
|ϕ|2 r dr dα ;
1 + a 2 β02
ω
this in turn yields the sought result.
It is clear from Equation (5) that the spectral threshold of h(0) cannot be lower
than that of −ωD . It has been shown in [7] that the inequality is sharp,
(8)
E := inf σ (h(0)) > inf σ −ωD ,
whenever ω is not rotationally symmetric. This follows, by the way, also from our
Lemma 2(b) which will be proved below.
Our aim is to show that this quantity determines the spectral threshold of our
original Hamiltonian, in other words, E = inf σ (H0 ). To this end let us denote by
f the real-valued eigenfunction of h(0) associated with the eigenvalue E = ε1 (0),
i.e.
h(0)f = −ωD f − β02 ∂2α f = Ef .
(9)
Then we can make the following claim.
LEMMA 2. Let f be given by Equation (9). Then
(a) f is strictly positive in ω.
(b) ω |fτ′ |2 dt = ω |∂α f |2 dt > 0 provided ω is not rotationally symmetric.
Proof. To prove the positivity of f it is enough to show that the semigroup
e−t h(0) is positivity improving for all t > 0, see [14, Theorem XIII.44], i.e., we have
to show that e−t h(0) maps every positive function in ω into a strictly positive function in ω. Since −ωD commutes with ∂2α , we get
ω
2 2
e−t h(0) = et D et β0 ∂α .
2 2
However, it follows easily from [14, Theorem XIII.50] that et β0 ∂α is positivity preserving for all t > 0, i.e., it maps every positive function into a positive function.
ω
Now note that since −ωD has a strictly positive ground state, et D is positivity
improving for all t > 0 by [14, Theorem XIII.44]. Hence given a positive function
2 2
g in ω, we know that et β0 ∂α g is positive, which means that e−t h(0) g is strictly positive; this proves the first statement of the Lemma.
188
P. EXNER AND H. KOVAŘÍK
The second statement is an immediate consequence of the first one. Let B be the
biggest circle centred at the origin, such that B ⊂ ω. Denote its complement in ω
by B c . By assumption we know that B c = ∅. Since f satisfies Dirichlet boundary
conditions on ∂ω and is strictly positive inside ω, it follows that |∂α f | is strictly
positive in almost every point of B c ∩ ∂ω, where ∂ω is not a part of a circle centred at the origin. This “non-circular” part is, of course, a positive measure set,
hence using the differentiability of f we can find a neighbourhood of B c ∩ ∂ω with
a positive Lebesgue measure on which |∂α f | > 0.
Remark. The first statement of Lemma 2 also follows from [10, Theorem. 8.38].
Now we are able to determine the spectrum of the free operator.
THEOREM 1. The spectrum of H0 is purely absolutely continuous and covers the
half-line [E, ∞), where E is the lowest eigenvalue of h(0).
Proof. From Equation (4) and Lemma 1 we know that the spectrum of H0 is
absolutely continuous and that [E, ∞) ⊂ σ (H0 ). It remains to show that
(−∞, E) ∩ σ (H0 ) = ∅ .
(10)
Using the fact that the ground-state eigenfunction f is strictly positive in ω, we
can decompose any ψ ∈ C0∞ (ω) as
ψ(s, t) = f (t)ϕ(s, t) .
(11)
We use the fact that f is real-valued and integrate by parts to get
Q0 [ψ] − E ψ2 =
f 2 |∇t ϕ|2 − (ωD f )f |ϕ|2 + f 2 |∂s ϕ|2 +
R×ω
+β0 f ∂α f (∂s ϕ̄ ϕ + ϕ̄ ∂s ϕ) + β0 f 2 (∂s ϕ̄ ∂α ϕ + ∂α ϕ̄ ∂s ϕ) +
+β02 f 2 |∂α ϕ|2 − β02 (∂2α f )f |ϕ|2 − E f 2 |ϕ|2 ds dt .
Since
R
(∂s ϕ̄ ϕ + ϕ̄ ∂s ϕ) ds = 0
and
−ωD f − β02 ∂2α f − E f = 0,
see (9), we finally obtain
2
Q0 [ψ] − E ψ =
R×ω
This implies Equation (10).
f 2 |∇t ϕ|2 + |∂s ϕ + β0 ϕτ′ |2 ds dt 0 .
189
SPECTRUM OF THE SCHRÖDINGER OPERATOR
4. Local Perturbations of the Twisting
After analyzing the “free” case, where the twisting velocity θ̇ was constant, we
want to look now what will happen if the translation invariance of our tube is
broken. We will suppose that the velocity of the twisting is given by
θ̇ (s) = β0 − β(s) ,
(12)
where β(·) is a bounded function such that supp β ⊂ [−s0 , s0 ] for some s0 > 0. Let
β denote the corresponding tube being defined by
β := L(R × ω) ,
where L refers to the twisting obtained by integration of (12). We use the symbol
Hβ for the Dirichlet Laplacian on L2 (β ) and
|∇ψ|2 ,
(13)
Qβ [ψ] :=
β
will be the associated quadratic form with the form domain D(Qβ ) = H01 (β ).
Since the support of the perturbation β(s) is compact, it is straightforward to
check that
σess (Hβ ) = σess (H0 ) = [E, ∞) .
(14)
Our main result says that if the tube twisting is locally slowed down in the mean,
the discrete spectrum of Hβ is non-empty.
THEOREM 2. Assume that ω is not rotationally symmetric and that
s0
(θ̇ 2 (s) − β02 )ds < 0 ,
(15)
−s0
where θ̇ (·) is given by Equation (12). Then the operator Hβ has at least one eigenvalue of finite multiplicity below the threshold of the essential spectrum.
Proof. Following the idea of [11] we start constructing a trial function from a
transverse eigenfunction corresponding to the bottom of the essential spectrum.
Given δ > 0 we put δ (s, t) = f (t) ϕ(s), where
 δ (s +s)
if s −s0 ,
e 0
ϕ(s) = 1
(16)
if −s0 s s0 ,
 −δ (s−s0 )
e
if s s0 .
It is easy to see that δ ∈ D(Qβ ). A straightforward calculation then gives
s0
(θ̇ 2 (s) − β02 )ds
Qβ [δ ] − E δ 2 = δ f 2L2 (ω) − fτ′ 2L2 (ω)
−s0
190
P. EXNER AND H. KOVAŘÍK
and
δ 2 = (δ −1 + 2s0 ) f 2L2 (ω) .
For δ → 0 we then get
fτ′ 2L2 (ω) s0
Qβ [δ ] − E δ 2
=
δ
(θ̇ 2 (s) − β02 ) ds + O(δ 2 ) .
δ 2
f 2L2 (ω) −s0
Thus in view of Lemma 2(b) it is sufficient to choose δ small enough to achieve
Qβ [δ ] − E δ 2
<0
δ 2
and the claim of the theorem follows.
Validity of the above result can be extended also to the critical case when the
integral in Equation (15) vanishes, however, we need a somewhat stronger assumption on the regularity of θ̇ . We also have to suppose that the twisting is “not fully
reverted” by the perturbation.
THEOREM 3. Assume that ω is not rotationally symmetric and let θ̇ (·) be given by
Equation (12). Suppose in addition that θ̇ (s) + β0 > 0 holds for |s| ≤ s0 , and that θ̈
exists and is of the class L2 ([−s0 , s0 ]). Let
s0
(θ̇ 2 (s) − β02 )ds = 0 ;
(17)
−s0
then the operator Hβ has at least one eigenvalue of finite multiplicity below the
threshold of the essential spectrum.
Proof. Following again the idea of [11] we improve the trial function used in the
proof of Theorem 2 by a deformation in the central region,
δ,γ (s, t) := f (t) ϕγ (s) ,
where
 δ (s +s)
if s −s0 ,
e 0
ϕγ (s) = 1 + γ (β0 − θ̇ (s)) if −s0 s s0 ,
 −δ (s−s0 )
e
if s s0 .
(18)
with γ > 0. Similarly as in the proof of Theorem 2 one can check that
ϕγ2 (fτ′ )2 θ̇ 2 (s) − β02 + f 2 (ϕγ′ )2 ds dt .
Qβ [δ,γ ] − E δ,γ 2 =
R×ω
Using the assumptions of the theorem we find that the integrals appearing in the
last expression behave as
s0
s0
2 θ̇ (s) − β0 θ̇ (s) + β0 ds + O(γ 2 ) ,
ϕγ2 θ̇ 2 (s) − β02 ds = −2γ
−s0
−s0
SPECTRUM OF THE SCHRÖDINGER OPERATOR
191
and
R
(ϕγ′ )2 ds = δ + γ 2
s0
−s0
2
θ̈ (s) ds = O(γ 2 ) + O(δ) .
as γ , δ → 0; the last two equations then give
fτ′ 2L2 (ω) s0 2 Qβ [δ,γ ] − E δ,γ 2
θ̇ (s) − β0 θ̇ (s) + β0 ds +
=
−2
γ
δ
2
2
δ,γ f L2 (ω) −s0
+δ O(γ 2 ) + O(δ 2 ) .
√
Setting now γ = δ we see that it is enough to make δ small enough to get
Qβ [δ,γ ] − E δ,γ 2
<0,
δ,γ 2
which concludes the proof.
Acknowledgements
The research has been partially supported by Czech Academy of Sciences and its
Grant Agency within the projects IRP AV0Z10480505 and A100480501, and by
DAAD within the project D-CZ 5/05-06.
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and three dimensions. Rev. Math. Phys. 7, 73–102 (1995)
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preparation)
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(2004)
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10. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order.
Springer, Berlin Heidelberg New York (1998)
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(1992)
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12. Kato, T.: Perturbation theory for linear operators. Springer, Berlin Heidelberg New
York (1966)
13. Londergan, J.T., Carini, J.P., Murdock, D.P.: Binding and scattering in two-dimensional
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Chapter 3
Metric trees
3.1
Weak coupling behaviour
Published in SIAM J. Math. Anal. 39 (2007) 1135–1149.
143
Weakly coupled Schrödinger operators on
regular metric trees
Hynek Kovařı́k
∗
Institute of Analysis, Dynamics and Modeling, Universität Stuttgart,
PF 80 11 40, D-70569 Stuttgart, Germany.
E-mail: [email protected]
Abstract
Spectral properties of the Schrödinger operator Aλ = −∆ + λV on regular
metric trees are studied. It is shown that as λ goes to zero the asymptotical
behavior of the negative eigenvalues of Aλ depends on the global structure
of the tree.
Mathematics Subject Classification: 34L40, 34B24, 34B45.
Key words: Schrödinger operator, Sturm-Liouville problems, metric trees.
1
Introduction
The spectrum of a Schrödinger operator
−∆ + λV ,
λ > 0,
in L2 (Rn ),
is given by the disjoint union of the essential spectrum σess (−∆ + λV ) and the
discrete spectrum σd (−∆+λV ). Under certain decay conditions on V at infinity
the essential spectrum covers the half-line [0, ∞) so that the discrete spectrum
consists of negative eigenvalues of finite multiplicity. It is a well known fact
that the behavior of these eigenvalues for small values of λ depend on the spacial
dimension n, [11]. Namely,
R for n < 3 the negative eigenvalues of −∆+λV appear
for any λ > 0, provided Rn V < 0, while for n ≥ 3 the negative spectrum of
−∆ + λV remains empty for λ small enough. Moreover for the lowest eigenvalue
−ε(λ) of −∆ + λV the following asymptotics hold true, see [11]:
n=1:
n=2:
ε(λ) ∼ λ2 ,
ε(λ) ∼ e
−λ−1
,
λ → 0,
λ → 0.
(1)
(2)
∗ Also on the leave from Nuclear Physics Institute, Academy of Sciences, 250 68 Řež near
Prague, Czech Republic.
1
In this paper we want to find the asymptotic behavior of ε(λ) for a Schrödinger
operator
Aλ = −∆ + λV , λ > 0 , in L2 (Γ)
defined on a regular metric tree Γ. Such a metric tree consists of the set of
vertices and the set of edges (branches), i.e. one dimensional intervals connecting
the vertices, see section 2 for details. Metric trees form a special subclass of
so called quantum graphs. The latter serve as mathematical models for nanotechnological devices that consist of connected very thin strips. The motion
of an electron in such a ”web” is then governed by the Schrödinger equation.
Therefore the study of Laplace and Schrödinger operators on these structures
has recently attracted a considerable attention, see e.g. [4, 5, 6, 7, 8, 9, 12, 13]
and references therein.
We are interested in the spectral behavior of Aλ when λ → 0. The intuitive
expectation is that this should depend on the rate of the growth, or branching,
of the tree Γ. In order to quantify this branching, we assign to Γ a so-called
global dimension d, see Definition 2 below. Roughly speaking it tells us how
fast the number of branches of Γ increases as a function of the distance from
its root. If the latter grows with the power d − 1 at infinity, then we say, in
analogy with the Euclidean spaces, that d is the global dimension of Γ. We use
the notation global in order to distinguish d from the local dimension, which is
of course one. Since d can be in general any real number larger or equal to one,
it is natural to ask how the weak coupling behavior looks like for non-integer
values of d and what is the condition on V under which the eigenvalues appear.
R Our main result says, see Theorem 3 and section 5.2, that if d ∈ [1, 2] and
V < 0, then Aλ possesses at least one negative eigenvalue for any λ > 0 and
Γ
for λ small enough this eigenvalue is unique and satisfies
ε(λ)
ε(λ)
2
∼ λ 2−d ,
∼ e
−λ−1
,
1 ≤ d < 2,
(3)
d = 2.
As expected, the faster the branching of Γ, i.e. larger d, the faster the eigenvalue tends to zero. The borderline is reached at d = 2 in which case ε(λ)
converges to zero faster than any power of λ similarly as in (2). Finally, if the
tree grows too fast, i.e. d > 2, then the discrete spectrum of Aλ generically
remains
empty for λ small enough, see sectionR5.3. Note also that the condition
R
V
<
0
is almost optimal in the sense that if Γ V > 0 then Aλ has no negative
Γ
eigenvalues for small λ as shown in Theorem 3b.
To study the operator Aλ we make use of the decomposition (5), Theorem
1, which was proved in [8, 9], see also [4]. In section 3.1 we introduce certain
auxiliary operators, whose eigenvalues will give us the estimate on ε(λ) from
above and from below. In order to establish (3) we find the asymptotics of
the lowest eigenvalues of the auxiliary operators, which are of the same order.
This is done in section 5.1. In addition, in section 4 we give some estimates on
the number of eigenvalues of the individual operators in the decomposition (5),
which might be of an independent interest as well.
2
Throughout the text we will employ the notation α := d − 1 and ν := 2−d
2 .
For a real-valued function f and a real non-integer number µ we will use the
shorthand
f |f |µ
.
f µ := sign f |f |µ =
|f |
Finally, given a self-adjoint operator T on a Hilbert space H we denote by
N− (T ; s) the number of eigenvalues, taking into account their multiplicities, of
T on the left of the point s. For s = 0 we will write N− (T ) instead of N− (T ; 0).
2
Preliminaries
We define a metric tree Γ with the root o following the construction given in [8].
Let V(Γ) be the set of vertices and E(Γ) be the set of edges of Γ. The distance
ρ(y, z) between any two points y, z ∈ Γ is defined in a natural way as the length
of the unique path connecting y and z. Consequently, |y| is equal to ρ(y, o). We
write y z if y lies on the unique simple path connecting o with z. For y z
we define
< y, z >:= {x ∈ Γ : y x z} .
If e =< y, z > is an edge, then y and z are its endpoints. For any vertex z its
generation Gen(z) is defined by
Gen(z) = #{x ∈ V : o ≺ x z} .
For an edge e ∈ E(Γ) we define its generation as the generation of the vertex,
from which e is emanating. The branching number b(z) of the vertex z is equal
to the number of edges emanating from z. We assume that b(z) > 1 for any
z 6= o and b(o) = 1.
Definition 1. A tree Γ is called regular if all the vertices of the same generation
have equal branching numbers and all the edges of the same generation have equal
length.
We denote by tk > 0 the distance between the root and the vertices of the k−th
generation and by bk ∈ N their corresponding branching number. For each
k ∈ N we define the so-called branching function gk : R+ → R+ by

if t < tk ,
 0
1
if tk ≤ t < tk+1 ,
gk (t) :=

bk+1 bk+2 · · · bn if tn ≤ t < tn+1 , k < n ,
and
g0 (t) := b0 b1 · · · bn
tn ≤ t < tn+1 .
It follows directly from the definition that
g0 (t) = #{x ∈ Γ : |x| = t} .
3
Obviously g0 (·) is a non-decreasing function and the rate of growth of g0 determines the rate of growth of the tree Γ. In particular, if one denotes by
Γ(t) := {x ∈ Γ : |x| ≤ t}, the “ball” of radius t, then g0 tells us how fast the
surface of Γ(t) grows with t. This motivates the following
Definition 2. If there exist positive constants a− , a+ and T0 , such that for all
t ≥ T0 the inequalities
g0 (t)
a− ≤ d−1 ≤ a+
t
hold true, then we say that d is the global dimension of the tree Γ.
We note that in the case of the so-called homogeneous metric trees treated
in [12] the function g0 (t) grows faster than any power of t. Formally, this
corresponds to d = ∞ in the above definition. From now on we will work under
the assumption that d < ∞.
3
Schrödinger operators on Γ
We will consider potential functions V which satisfy the
Assumption A. V : R+ → R is measurable, bounded and limt→∞ V (t) = 0.
For a given function V which satisfies the Assumption A we define the Schrödinger
operator Aλ as the self-adjoint operator in L2 (Γ) associated with the closed
quadratic form
Z
|u′ |2 + λV (|x|) |u|2 dx ,
Qλ [u] :=
Γ
with the form domain D(Q) = H 1 (Γ) consisting of all continuous functions u
such that u ∈ H 1 (e) on each edge e ∈ E(Γ) and
Z
|u′ |2 + |u|2 dx < ∞ .
Γ
The domain of Aλ consists of all continuous functions u such that u′ (o) = 0 ,
u ∈ H 2 (e) for each e ∈ E(Γ) and such that at each vertex z ∈ V(Γ) \ {o} the
matching conditions
u− (z) = u1 (z) = · · · = ub(z) (z) ,
u′1 (z) + · · · + u′b(z) (z) = u′− (z)
(4)
are satisfied, where u− denotes the restriction of u on the edge terminating in
z and uj , j = 1, ..., b(z) denote respectively the restrictions of u on the edges
emanating from z, see [8] for details. Notice that Aλ satisfies the Neumann
boundary condition at the root o.
The following result by Naimark and Solomyak, see [8, 9], also established
by Carlson in [4], makes it possible to reduce the spectral analysis of Aλ to the
analysis of one dimensional Schrödinger operators in weighted L2 (R+ ) spaces:
4
Theorem 1. Let V be measurable and bounded and suppose that Γ is regular.
Then Aλ is unitarily equivalent to the following orthogonal sum of operators:
Aλ ∼ Aλ,0 ⊕
[b ...b
∞
X
k=1
[b ...bk−1 (bk −1)]
1
⊕ Aλ,k
.
(5)
(b −1)]
1
k−1 k
Here the symbol Aλ,k
means that the operator Aλ,k enters the orthogonal sum [b1 ...bk−1 (bk − 1)] times. For each k ∈ N the corresponding selfadjoint operator Aλ,k acts in L2 ((tk , ∞), gk ) and is associated with the closed
quadratic form
Z ∞
Qk [f ] =
|f ′ |2 + λV (t) |f |2 gk (t) dt ,
tk
whose form domain is given by the the weighted Sobolev space D(Qk ) =
H01 ((tk , ∞), gk ) which consists of all functions f such that
Z ∞
|f ′ |2 + |f |2 gk (t) dt < ∞ , f (tk ) = 0 .
tk
The operator Aλ,0 acts in the weighted space L2 (R+ , g0 ) and is associated with
the closed form
Z ∞
|f ′ |2 + λV (t) |f |2 g0 (t) dt ,
Q0 [f ] =
0
with the form domain D(Q0 ) = H 1 (R+ , g0 ) which consists of all functions f
such that
Z ∞
|f ′ |2 + |f |2 g0 (t) dt < ∞ ,
0
see also [13].
3.1
Auxiliary operators
Let d be the global dimension of Γ. Definition 2 implies that there exist positive
constants b− and b+ , such that
b− (1 + t)α =: gk− (t) ≤ gk (t) ≤ gk+ (t) := b+ (1 + t)α ,
Now assume that the Rayleigh quotient
R∞
|f ′ |2 + λV (t) |f |2 gk (t) dt
tk
R∞
|f |2 gk (t) dt
tk
t ∈ [tk , ∞) .
(6)
of the operator Aλ,k , k ≥ 0 is negative for some f ∈ D(Qk ). From (6) follows
that
R∞
R∞
|f ′ |2 + λV (t) |f |2 gk (t) dt
|f ′ |2 + λVk− (t) |f |2 (1 + t)α dt
tk
tk
R∞
R∞
≤
|f |2 (1 + t)α dt
|f |2 gk (t) dt
tk
tk
R∞
|f ′ |2 + λVk+ (t) |f |2 (1 + t)α dt
R∞
≤ tk
,
(7)
|f |2 (1 + t)α dt
tk
5
where
Vk− (t) :=
gk (t)
V (t) ,
gk− (t)
Vk+ (t) :=
gk (t)
V (t) .
gk+ (t)
It is thus natural to introduce the auxiliary operators A±
λ,k acting in the Hilbert
2
α
space L ((tk , ∞), (1 + t) ) and associated with the quadratic forms
Z ∞
Q±
[f
]
=
|f ′ |2 + λVk± (t) |f |2 (1 + t)α dt , f ∈ D(Qk ) , k ∈ N0 . (8)
k
tk
The variational principle, see e.g. [3], and (7) thus imply that
−
N− (A+
λ,k ; s) ≤ N− (Aλ,k ; s) ≤ N− (Aλ,k ; s) ,
s ≤ 0, k ∈ N0 .
(9)
Let En,k (λ) be the non-decreasing sequence of negative eigenvalues of the op±
erators Aλ,k and let En,k
(λ) be the analogous sequences corresponding to the
±
operators Aλ,k respectively. In all these sequences each eigenvalue occurs according to its multiplicity. Relation (9) and variational principle then yield
−
+
En,k
(λ) ≤ En,k (λ) ≤ En,k
(λ) ,
and
k ∈ N0 , n ∈ N ,
+
inf σess (A−
λ,k ) ≤ inf σess (Aλ,k ) ≤ inf σess (Aλ,k ) ,
Next we introduce the transformation U by
k ∈ N0
(10)
(11)
(U f )(t) = (1 + t)α/2 f (t) =: ϕ(t) ,
which maps L2 ((tk , ∞), (1 + t)α ) unitarily onto L2 ((tk , ∞)). We thus get
Lemma 1. Let V satisfy the assumptions of Theorem 1. Then
(i) For each k ∈ N the operators A±
λ,k are unitarily equivalent to the self±
adjoint operators Bλ,k
in L2 ((tk , ∞)), which act as
(d − 1)(d − 3)
±
Bλ,k
ϕ (t) = −ϕ′′ (t) +
ϕ(t) + λVk± (t) ϕ(t) ,
4(1 + t)2
(12)
and whose domains consist of all functions ϕ ∈ H 2 ((tk , ∞)) such that
ϕ(tk ) = 0 .
±
2
(ii) A±
λ,0 are unitarily equivalent to the self-adjoint operators Bλ,0 in L (R+ ),
acting as
(d − 1)(d − 3)
±
Bλ,0
ϕ (t) = −ϕ′′ (t) +
ϕ(t) + λ V0± (t) ϕ(t) ,
(13)
4(1 + t)2
with the domain that consists of all ϕ ∈ H 2 (R+ ) such that
ϕ′ (0) =
6
d−1
ϕ(0) .
2
(14)
Proof. For each k ∈ N0 we have
±
−1
Bλ,k
= U A±
,
λ,k U
kf kL2 ((tk ,∞), (1+t)α ) = kU f kL2((tk ,∞)) .
The statement of the Lemma then follows by a direct calculation keeping in mind
′
that the functions f from the domain of the operators A±
λ,0 satisfy f (0) = 0.
Remark 1. If V satisfies assumption A, then the inequalities (11) and standard arguments from the spectral theory of Schrödinger operators, see e.g. [10,
Chap.13.4], imply that
+
inf σess (A−
λ,k ) = inf σess (Aλ,k ) = inf σess (Aλ,k ) = 0 ,
∀ k ∈ N0 .
Moreover, constructing suitable Weyl sequences for the operators Aλ,k in the
similar way as it was done in [13] for the Laplace operator, one can easily show
that
σess (Aλ,k ) = [0, ∞) , ∀ k ∈ N0 .
(15)
4
Number of bound states
From Theorem 1 and equation (15) we can see that if V satisfies assumption A
then
σess (Aλ ) = [0, ∞) .
(16)
In order to analyze the discrete spectrum of Aλ we first study the number of
bound states of the individual operators in the decomposition (5).
We start by proving an auxiliary Proposition. Given a real valued measurable bounded function Ṽ we consider the self-adjoint operator B̃λ acting in
L2 (R+ ) as
(d − 1)(d − 3)
ϕ(t) + λṼ (t) ϕ(t)
(17)
B̃λ ϕ (t) = −ϕ′′ (t) +
4 t2
with the Dirichlet boundary condition at zero. This operator is associated with
the closure of the quadratic form
Z (d − 1)(d − 3)
2
2
|ϕ′ (t)|2 +
dt
|ϕ(t)|
+
λ
Ṽ
(t)
|ϕ(t)|
4 t2
R+
defined on C0∞ (R+ ).
1. Let d ∈ [1, 2). Assume that Ṽ satisfies assumption A and that
RProposition
∞
t
|
Ṽ
(t)|
dt
< ∞. Then
0
Z ∞
N− (B̃λ ) ≤ λ K̃(d)
t |Ṽ (t)| dt ,
(18)
0
where
K̃(d) =
π
.
2 sin(νπ)Γ(1 − ν)Γ(1 + ν)
7
Proof. We write
B̃λ = B̃0 + λ Ṽ ,
B̃0 := −
(d − 1)(d − 3)
d2
+
.
d t2
4t2
Moreover, without loss of generality we may assume that Ṽ is negative and
continuous. The statement for the general case then follows by a standard
approximation argument.
By the Birman-Schwinger principle, see e.g.[3], the number of eigenvalues of
B̃λ to the left of the point −κ2 then does not exceed the trace of the operator
λ|Ṽ |1/2 (B̃0 + κ2 )−1 |Ṽ |1/2 .
The integral kernel G̃(t, t′ , κ) of the operator (B̃0 + κ2 )−1 can be calculated by
using the Sturm-Liouville theory. We get
 πi
 4 v1 (t, κ) v2 (t′ , κ) t ≥ t′
′
,
(19)
G̃(t, t , κ) =
 πi
′
′
v
(t
,
κ)
v
(t,
κ)
t
<
t
1
2
4
with
√
v1 (t, κ) =
t Hν(1) (iκt) ,
√
√ (1)
t Hν (iκt) + t Hν(2) (iκt) ,
v2 (t, κ) =
(1)
(2)
where Hν resp. HνR denote Hankel’s functions of the first resp. second kind,
∞
see e.g. [14]. Since 0 t |Ṽ (t)| dt < ∞, we can pass to the limit κ → 0 in the
corresponding integral, using the Lebesgue dominated convergence theorem, and
calculate the trace to get
Z ∞
Z ∞
N− (B̃λ ) ≤ λ
|Ṽ (t)| |G̃(t, t, 0)| dt = λ K̃(d)
t |Ṽ (t)| dt .
(20)
0
0
Here we have used the fact that G̃(t, t, κ) → t K̃(d) pointwise as κ → 0, which
follows from the asymptotic behavior of the Hankel functions at zero, see e.g.
[1].
Remark 2. For d = 1 we have K̃(1) = 1 and (18) gives the well known
Bargmann inequality, [2]. On the other hand, K̃(d) diverges as d → 2. This
2
is expected because the operator − ddt2 − 4t12 + λV with Dirichlet b.c. at zero
does have at least one negative eigenvalue for any λ > 0 if the integral of V is
negative.
Armed with Proposition 1 we can prove
1. Let 1 ≤ d < 2. Assume that V satisfies assumption A and that
RCorollary
∞
t
|V
(t)|
dt
< ∞. Then
0
Z ∞
N− (Aλ,0 ) ≤ 1 + λ K(d)
|V (t)| g0 (t) t2−d dt.
(21)
0
8
Proof. We introduce the operator AD
λ,0 , which is associated with the quadratic
form
Z ∞
1
|f ′ |2 + λV (t) |f |2 g0 (t) dt , D(QD
QD
[f
]
:=
0 ) = H0 (R+ , g0 ) ,
0
0
where H01 (R+ , g0 ) := {f ∈ H 1 (R+ , g0 ), f (0) = 0}. First we observe that
a td−1 ≤ g0 (t) ,
t ∈ R+
for a suitable a > 0. We can thus mimic the analysis of Section 3.1 and define
the operator Ãλ acting in L2 (R+ , td−1 ) associated with the quadratic form
Z ∞
(22)
|f ′ |2 + λṼ (t) |f |2 td−1 dt , f ∈ D(Q)
Q̃[f ] =
0
H01 ((R+ ), td−1 )
where D(Q) =
and Ṽ (t) :=
ments of Section 3.1 we claim that
g0 (t)
a td−1
V (t) . Repeating the argu-
N− (AD
λ,0 ) ≤ N− (Ãλ )
and that Ãλ is unitarily equivalent to B̃λ by means of the transformation
Ũ f (t) = t(d−1)/2 f (t), which maps L2 (R+ , td−1 ) unitarily onto L2 (R+ ). Since
the co-dimension of H01 (R+ , g0 ) in H 1 (R+ , g0 ) is equal to one, the variational
principle gives
N− (Aλ,0 ) ≤ 1 + N− (AD
λ,0 ) ≤ 1 + N− (Ãλ ) = 1 + N− (B̃λ ) .
Application of Proposition 1 with K̃(d) = a K(d) concludes the proof.
Corollary
2. Let 1 ≤ d < 2. Let V satisfy assumption A and assume that
R∞
t
|V
(t)|
dt
< ∞ . Then there exists λc > 0, so that for λ ∈ [0, λc ] the discrete
0
spectra of the operators Aλ,k , k ≥ 1 are empty. In particular we have
σd (Aλ ) = σd (Aλ,0 ) ,
0 ≤ λ ≤ λc ,
(23)
where the multiplicities of the eigenvalues are taken into account.
Proof. In view of Lemma 1 it suffices to show that the discrete spectra of the
−
operators Bλ,k
for k ≥ 1 are empty provided λ is small enough. Since (d−1)(d−
3) ≤ 0, the following inequality holds true in the sense of quadratic forms:
−
Bλ,k
≥ Bλ,k := −
(d − 1)(d − 3)
d2
+
+ λVk− (t) ,
dt2
4(t − tk )2
(24)
where Bλ,k acts in L2 ((tk , ∞)) with Dirichlet boundary conditions at tk . A
simple translation s = t − tk then shows that Bλ,k is unitarily equivalent to the
operator
d2
(d − 1)(d − 3)
− 2+
+ λVk− (s + tk ) in L2 (R+ )
ds
4s2
R∞
which is defined in the similar way as the operator B̃λ in (17). Since 0 s |Vk− (s+
tk )| ds is uniformly bounded with respect to k, it follows from Proposition 1 that
for λ small enough we have N− (Bλ,k ) = 0 for all k ≥ 1. In view of (24) this
concludes the proof.
9
5
Weak coupling
5.1
The case 1 ≤ d < 2
In this section we will show that if d ∈ [1, 2) and V is attractive in certain sense,
then the operator Aλ possesses at least one negative eigenvalue for any λ > 0.
Since for small values of λ the discrete spectra of Aλ and Aλ,0 coincide, see
Corollary 2, we will focus on the operator Aλ,0 only. More exactly, in view of
±
(10), we will study the operators Bλ,0
. Clearly we have
±
Bλ,0
= B0 + λ V0± ,
B0 := −
(d − 1)(d − 3)
d2
+
,
d t2
4(1 + t)2
with the boundary condition v ′ (0) = d−1
2 v(0). Note that, by Lemma 1, the
operator B0 is non-negative. We shall first calculate the Green function of B0
at a point −κ2 , κ > 0, using the Sturm-Liouville theory again. In the same
manner as in the previous section we obtain

π
′
′
 4iβ(κ) v1 (t, κ) v2 (t , κ) t ≥ t
′
,
(25)
G(t, t , κ) :=

π
′
′
v
(t
,
κ)
v
(t,
κ)
t
<
t
1
2
4iβ(κ)
where
v1 (t, κ) =
v2 (t, κ) =
√
1 + t Hν(1) (iκ(1 + t)) ,
√
1 + t Hν(1) (iκ(1 + t)) − β(κ) Hν(2) (iκ(1 + t)) ,
(1)
β(κ)
=
Hν−1 (iκ)
(2)
Hν−1 (iκ)
.
Consider a function W which satisfies assumption A. According to the
Birman-Schwinger principle the operator B0 + λW has an eigenvalue −κ2 if and
only if the operator
K(κ) := |W |1/2 (B0 + κ2 )−1 W 1/2
has eigenvalue −λ−1 . The integral kernel of K(κ) is equal to
K(t, t′ , κ) = |W (t)|1/2 G(t, t′ , κ) (W (t′ ))1/2 .
We will use the decomposition
K(t, t′ , κ) = L(t, t′ , κ) + M (t, t′ , κ) ,
with
L(t, t′ , κ) :=
1
π 22ν−1 κ−2ν
|W (t)|1/2 [(1 + t)(1 + t′ )]−ν+ 2 W (t′ )1/2 ,
(Γ(1 − ν))2 sin(νπ)
10
and denote by L(κ) and M (κ) the integral operators with the kernels L(t, t′ , κ)
and M (t, t′ , κ) respectively. Furthermore, we denote by M (0) the integral operator with the kernel
ν sign (t−t′ )
1
1+t
M (t, t′ , 0) := CM (ν) |W (t)|W (t′ ) (1 + t)(1 + t′ ) 2
1 + t′
where
CM (ν) := −
π
.
2 sin(νπ)Γ(1 − ν)Γ(1 + ν)
Lemma 2 in the Appendix says that M (κ) converges in the Hilbert-Schmidt
norm to the operator M (0) as κ → 0, provided W decays fast enough at infinity.
This allows us to prove
R∞
Theorem 2. Assume that W satisfies A and that 0 (1 + t)3−d |W (t)| dt < ∞,
where 1 ≤ d < 2. Then the following statements hold true.
(a) If
Z
∞
W (t) (1 + t)d−1 dt < 0 ,
0
then the operator B0 + λ W has at least one negative eigenvalue for all
λ > 0. For λ small enough this eigenvalue, denoted by E(λ), is unique
and satisfies
Z ∞
2−d
d−1
2
2
W (t) (1 + t)
dt + O(λ ) ,
(26)
= C(ν) λ
(E(λ))
0
where
C(ν) =
(b) If
Z
∞
π 22ν−1
.
(Γ(1 − ν))2 sin(νπ)
W (t) (1 + t)d−1 dt > 0 ,
0
then the operator B0 + λ W has no negative eigenvalues for λ positive and
small enough.
Proof. Part (a).
The operator B0 + λ W has eigenvalue E = −κ2 if and only if the operator
λK(κ) = λM (κ) + λL(κ)
has an eigenvalue −1 for certain κ(λ). On the other hand, Lemma 1 and (9)
imply that
g0
N− B0 + λ + V ≤ N− (Aλ,0 ) .
g0
11
The uniqueness of E, and so of κ(λ), for λ small enough thus follows from (21)
g+
by taking V = g00 W . Next we note that by Lemma 2 for λ small we have
λ kM (κ)k < 1 and
−1
(I + λK(κ))
−1
(I + λ M (κ))−1 .
= I + λ(I + λM (κ))−1 L(κ)
Hence λK(κ) has an eigenvalue −1 if and only if λ(I + λM (κ))−1 L(κ) has an
eigenvalue −1. Since λ(I + λM (κ))−1 L(κ) is of rank one we get the equation
for κ(λ) in the form
tr λ(I + λM (κ(λ))−1 L(κ(λ)) = −1 .
(27)
Using the decomposition
(I + λM (κ))−1 = I − λM (0) − λ(M (κ) − M (0)) + λ2 M 2 (κ)(I + λM (κ))−1
we obtain
tr λ(I + λM (κ))−1 L(κ)
1
1
= λ C(ν)κ−2ν |W (t)|1/2 (1 + t)−ν+ 2 , (I + λM (κ))−1 W (t)1/2 (1 + t)−ν+ 2
Z ∞
W (t) (1 + t)d−1 dt + O(λ2 ) .
= C(ν) κ−2ν λ
0
It thus follows from (27) that
Z
E ν (λ) = −κ2ν (λ) = C(ν) λ
∞
0
W (t) (1 + t)d−1 dt + O(λ2 ) .
(28)
To finish the proof of the part (a) of the Theorem we mimic the argument used
in [11] and notice that if (ϕ, (B0 + λW ) ϕ) < 0, then (ϕ, W ϕ) < 0, since B0 is
non-negative, and therefore (ϕ, (B0 + λ̃W ) ϕ) < 0 if λ < λ̃. So if B0 + λW has
a negative eigenvalue for λ small enough, then, by the variational principle, it
has at least one negative eigenvalue for all λ positive.
Part (b). From the proof of part (a) it can be easily seen that if
Z ∞
W (t) (1 + t)d−1 dt > 0 ,
0
then tr λ(I + λM (κ))−1 L(κ) is positive for λ small and therefore K(κ) cannot
have an eigenvalue −1.
Remark 3. Note that if
W0 :=
Z
′
1−ν
W (t) W (t )(1 + t)
′ 1−ν
(1 + t )
R2+
12
1+t
1 + t′
ν sign (t−t′ )
dt dt′ < 0 ,
then the operator B0 + λW has a negative eigenvalue for λ small, positive or
negative, also in the critical case when
Z ∞
W (t) (1 + t)d−1 dt = 0 .
0
Moreover, it follows from the proof of Theorem 2 that this eigenvalue then
satisfies
E ν (λ) = C(ν) −λ2 CM (ν) W0 + o (λ2 ) , λ → 0 .
(29)
As an immediate consequence of Theorem 2 and inequalities (10) we get
R∞
Theorem 3. Let V satisfy assumption A and let 0 (1 + t)3−d |V (t)| dt < ∞,
where 1 ≤ d < 2. Then the following statements hold true.
(a) If
Z
∞
V (t) g0 (t) dt =
0
Z
V (|x|) dx < 0 ,
Γ
then the operator Aλ has at least one negative eigenvalue E1,0 (λ) for all
λ > 0. For λ small enough this eigenvalue is unique and satisfies
2
2
Z
2−d
Z
2−d
V (|x|) dx
C1 λ
V (|x|) dx
≤ |E1,0 (λ)| ≤ C2 λ
(30)
Γ
Γ
for suitable positive constants C1 and C2 .
(b) If
Z
∞
V (t) g0 (t) dt =
0
Z
V (|x|) dx > 0 ,
Γ
then the discrete spectrum of Aλ is empty for λ positive and small enough.
Proof. Part (a). From (10) we get
−
+
E1,0
(λ) ≤ E1,0 (λ) ≤ E1,0
(λ) .
±
±
Moreover, by Lemma 1 E1,0
(λ) are the lowest eigenvalues of operators Bλ,0
.
The existence and uniqueness of E1,0 thus follows from part (a) of Theorem 2
applied with W (t) = V0+ (t) and W (t) = V0− (t) respectively. At the same time,
equation (26) implies (30).
Similarly, part (b) of the statement follows immediately from Lemma 1 and
part (b) of Theorem 2 applied with W (t) = V0− (t).
Remark 4. We note that the strong coupling behavior of Aλ is, on the contrary to (30), typically one-dimensional, i.e. determined by the local dimension
of Γ. Namely, if V is continuous and compactly supported, then the standard
Dirichlet-Neumann bracketing technique shows that the Weyl asymptotic formula
Z
X
1
1
|Ej |γ = Lcl
|V |γ+ 2 dx, γ ≥ 0
lim λ−γ− 2
γ,1
λ→∞
Γ
j
holds true, where Ej are the negative eigenvalues of Aλ and Lcl
γ,1 =
13
Γ(γ+1)
√
.
2 π Γ(γ+3/2)
Remark 5. Notice that our result qualitatively agrees with the precise asymptotic formula for ε(λ) on branching graphs with one vertex and finitely many
edges which was found in [5]. Such graphs correspond to d = 1 in our setting.
5.2
The case d = 2
For d = 2 one can mimic the above procedure replacing the Hankel functions
(1,2)
(1,2)
Hν
by H0 . The latter have a logarithmic singularity at zero and therefore
it turns out that the lowest eigenvalue of Aλ then converges to zero exponentially
fast. Indeed, here instead of (26) one obtains
−1
E(λ) ∼ −e−λ
,
λ→0
as for the two-dimensional Schrödinger operator, see [11]. Since the analysis of
this case is completely analogous to the previous one, we skip it.
5.3
The case d > 2
Here we will to show, under some assumptions on V , that for d > 2 and λ small
enough
the discrete spectrum of Aλ remains empty no matter what the sign of
R
V
is.
Γ
Proposition 2. Let d > 2 and let V satisfy assumption A. If V ∈ L∞ (R+ ) ∩
Lp/2 (R+ , g0 ) with p < d, then there exists λ0 > 0 such that the discrete spectrum
of Aλ is empty for all λ ∈ [0, λ0 ].
Proof. From the definition of the function gk it follows that
R∞
R∞
′ 2
2
′ 2
2
g0 (t) dt
gk (t) dt
tk |f | + λV |f |
tk |f | + λV |f |
R∞
R∞
=
.
2
2
tk |f | gk (t) dt
tk |f | g0 (t) dt
Since every function f ∈ H01 ((tk , ∞), gk ) can be extended by zero to a function
in H 1 (R+ , g0 ), the variational principle shows that
σd (Aλ,0 ) = ∅ =⇒ σd (Aλ,k ) = ∅
∀k ≥ 1.
Hence it suffices to prove the statement for the operator Aλ,0 , i.e. to show that
Aλ,0 is non-negative. Consider a function f ∈ D(Q0 ). Since f ∈ H 1 (R+ ), which
is continuously embedded in L∞ (R+ ), it follows that f → 0 at infinity and we
can write
Z ∞
f (t) = −
f ′ (s) ds .
t
In view of (6) we have g0−1 ∈ L1 (R+ ). Using Cauchy-Schwarz inequality we thus
find out that for any q > q0 , where q10 + d1 = 21 , the following estimate holds
14
true
Z
∞
0
≤
q
|f (t)| g0 (t) dt
Z
∞
0
≤ C(q)
∞
Z
t
Z
∞
0
q1
Z
≤
∞
0
∞
Z
′
|f (s)|ds
t
|f ′ (s)|2 g0 (s) ds
|f ′ (s)|2 g0 (s) ds
2q Z
∞
t
21
ds
g0 (s)
0
∞
2
|V | |f | g0 (t) dt
≤
≤
Z
∞
0
2
C (q)
p/2
|V |
Z
0
q2
g0 (t) dt
! q1
(31)
g0 (t) dt
p2 Z
1
q
∞
0
∞
g0 (t) dt
1q
,
with a constant C(q) independent of f . Take q such that
inequality and (31) then give
Z
q
′ 2
|f | g0 (t) dt
+ p1 = 21 . The Hölder
|f | g0 (t) dt
∞
Z
q
0
p/2
|V |
q2
g0 (t) dt
p2
,
which implies
Q0 [f ] ≥
Z
∞
0
"
′ 2
2
|f | g0 (t) dt 1 − λ C (q)
Z
0
∞
p/2
|V |
g0 (t) dt
p2 #
.
To show that the negative spectrum of Aλ,0 is empty it suffices to take λ small
enough so that Q0 [f ] ≥ 0.
Appendix
R∞
Lemma 2. Let W be bounded and assume that 0 (1 + t)1+2ν |W (t)| dt < ∞.
Then M (κ) converges in the Hilbert-Schmidt norm to the operator M (0) as
κ → 0.
Proof. We first notice that M (0) is Hilbert-Schmidt, since
Z ∞Z ∞
|M (t, t′ , 0)|2 dt dt′ < ∞
0
0
by assumption. We will also need the asymptotic behavior of the Bessel functions with purely imaginary argument near zero:
Jν (iκ(1 + t)) = eiπν/2 Iν (κ(1 + t)) ∼ eiπν/2
κν (1 + t)ν
,
2ν Γ(ν + 1)
κ(1 + t) → 0 , (32)
see [1, 14]. From the definition of Hankel’s functions we thus get
β(κ) =
J1−ν (iκ) − ei(1−ν)π Jν−1 (iκ)
→ −e−2iνπ ,
ei(ν−1)π Jν−1 (iκ) − J1−ν (iκ)
15
κ → 0.
This together with the asymptotics (32) implies
lim M (t, t′ , κ) = M (t, t′ , 0) .
(33)
κ→0
Now using the asymptotic behavior of Hankel’s functions at infinity, [1], we find
out that
′
′
′
1/2
G(t, t , κ) ∼ ((1 + t)(1 + t ))
′
e−κ(t+t ) − β(κ) e−κ|t−t |
,
κ (1 + t)(1 + t′ )1/2
κ2 (1+t)(1+t′ ) → ∞ .
Since |β(κ)| is bounded, we obtain the following estimates.
For κ2 (1 + t)(1 + t′ ) ≥ 1:
1/2
|K(t, t′ , κ)| , |L(t, t′ , κ)| ≤ C |W (t′ ) W (t)(1 + t)(1 + t′ )|
.
For κ2 (1 + t)(1 + t′ ) < 1:
i
h
1
|M (t, t′ , κ)| ≤ C ′ |W (t′ ) W (t) | 1 + ((1 + t)(1 + t′ ))ν+ 2 ,
where we have used (33). Note that the constants C and C ′ may be chosen independent of κ, which enables us to employ the Lebesgue dominated convergence
theorem to conclude that
Z
lim
|M (t, t′ , κ) − M (t, t′ , 0)|2 dt dt′ = 0 .
κ→0
R2+
Acknowledgement
The work has been partially supported by the Czech Academy of Sciences and
by DAAD within the project D-CZ 5/05-06.
References
[1] M. Abramowitz and I.A. Stegun: Handbook of Mathematical Functions,
National Bureau of Standards (1964).
[2] V. Bargmann: On the number of bound states in a central field of force,
Proc. Nat. Acad. Sci. U.S.A. 38 (1952) 961–966.
[3] M.S. Birman and M.Z. Solomyak: Schrödinger Operator. Estimates for
number of bound states as function-theoretical problem, Amer. Math. Soc.
Transl. (2) Vol. 150 (1992).
[4] R. Carlson, Nonclassical Sturm-Liouville problems and Schrödinger operators on radial trees, Elect. J. Diff. Equation 71 (2000).
16
[5] P. Exner: Weakly Coupled States on Branching Graphs, Letters in Math.
Phys. 38 (1996) 313–320.
[6] P. Kuchment: Quantum graphs: I. Some basic structures. Waves in Rand.
Media 14 (2004) 107–128.
[7] V. Kostrykin and R. Schrader: Kirchhoff’s rules for quantum wires.
J. Phys. A 32 (1999) 595–630.
[8] K. Naimark and M. Solomyak, Geometry of the Sobolev spaces on the
regular trees and Hardy’s inequalities, Russian Journal of Math. Phys. 8
(2001) 322–335.
[9] K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. 80 (2000) 690–724.
[10] M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV, Academic press, New York (1978).
[11] B. Simon, The Bound State of Weakly Coupled Schrödinger Operators in
One and Two Dimensions, Ann. of Physics 97 (1976) 279–288.
[12] A. Sobolev and M. Solomyak, Schrödinger operators on homogeneous metric trees: spectrum in gaps, Rev. Math. Phys. 14 (2002) 421–467.
[13] M. Solomyak, On the spectrum of the Laplacian on metric trees, Waves in
Rand. Media 14 (2004) S155–S171.
[14] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge
University Press (1958).
17
3.2
Weighted Lieb-Thirring inequalities
Preprint: www.mathematik/uni-stuttgart.de/preprints: 2007/004.
163
EIGENVALUE ESTIMATES FOR SCHRÖDINGER OPERATORS ON
METRIC TREES
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
Abstract. We consider Schrödinger operators on regular metric trees and prove LiebThirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the
bounds are valid in the endpoint case and reflect the correct order in the weak or strong
coupling limit.
1. Introduction
It is well known that the moments of negative eigenvalues of the Schrödinger operator
−∆ − V in L2 (Rd ) can be estimated in terms of the classical phase space volume. Namely,
the Lieb-Thirring inequality states that the bound
Z
γ+ d
V+ 2 dx
tr (−∆ − V )γ− ≤ Lγ,d
(1.1)
Rd
holds true for any potential V if and only if
1
γ ≥ if d = 1, γ > 0 if d = 2, γ ≥ 0 if d ≥ 3 .
(1.2)
2
Here x± := max{0, ±x} denotes the positive and negative part of x. Inequality (1.1) is due
to Lieb and Thirring [26] and, in the endpoint cases, to Cwikel [6], Lieb [24], Rozenblum
[30] and Weidl [34]. We refer to [23] and [15] for recent reviews on this topic.
Our main objective is to establish the analog of (1.1) for Schrödinger operators on metric
trees. A (rooted) metric tree Γ consists of a set of vertices and a set of edges, i.e., segments
of the real axis which connect the vertices. We assume that Γ has infinite height, that is, it
contains points at arbitrary large distance from the root. We define the Schrödinger operator
formally as
−∆N − V
in
L2 (Γ)
with Kirchhoff matching conditions at the vertices and a Neumann boundary condition at
the root of the tree.
Metric trees represent a special class of so called quantum graphs, which recently have attracted great interest; see, e.g., [3, 18, 20, 21] for extensive bibliographies about this subject.
Many works devoted to quantum graphs concern questions about self-adjoint extensions,
approximation by thin quantum wave guides and direct or inverse scattering properties of
the Laplace operator on graphs, see the references above and also [11, 22]. Various functional
inequalities for the Laplacian on metric trees have been established in [10, 27]. However,
much less attention has been paid, with the exception of [28], to the classical question of
finding appropriate estimates, similar to (1.1), on the discrete spectrum of Schrödinger operators on metric trees. As we shall see, the interplay between the spectral theory and the
mixed dimensonality of a tree makes this a fascinating problem.
Key words and phrases. Schrödinger operator, metric tree, eigenvalue estimate, Lieb-Thirring inequality,
Cwikel-Lieb-Rozenblum inequality.
c 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
1
2
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
Our main result concern regular metric trees, that is, trees which are symmetric with
respect to the distance from the root; see Subsection 2.1 for a precise definition. We shall
show that the validity of a suitable analog of (1.1) is characterized by the global branching
of the tree Γ. The latter is expressed by the branching function g0 (t) := #{x : |x| = t}
which counts the number of points of Γ as a function of the distance from the root. The
function g0 is clearly non-decreasing. Depending on its growth we may split the trees into
two classes according to whether the integral
Z ∞
dt
(1.3)
g0 (t)
0
is finite (transient trees) or infinite (recurrent trees). It turns out that in the former case,
the corresponding Lieb-Thirring inequality holds for all values γ ≥ 0. For γ = 0 this is
an estimate on the number of negative eigenvalues in terms of an integral of the potential,
usually called a Cwikel-Lieb-Rozenblum inequality. On the other hand, if the integral (1.3)
is infinite, then Lieb-Thirring inequalities do not hold for values of γ which are smaller than
some critical value γmin > 0. In order to determine the value of γmin we use the notion of the
global dimension of a metric tree, see Definition 2.5. This dimension is equal to d ≥ 1 if the
branching function g0 has a power-like growth at infinity with power d − 1. We emphasize
that in contrast to the Euclidean case, d need not be an integer.
For regular metric trees Γ with global dimension d and Schrödinger operators with symmetric potentials V we shall prove Lieb-Thirring inequalities of the form
Z
a
γ+ 1+a
(1.4)
V+ 2 g0d−1 dx , a ≥ 0 .
tr (−∆N − V )γ− ≤ C
Γ
The allowed values of γ are determined by the parameter a and by the global dimension d of
Γ, see Theorem 2.7. For a = 0 the weight in the integral on the right hand side disappears
and the inequality is very similar to its Euclidean version (1.1). Both sides then share the
same growth in the strong coupling limit, see Remark 2.10 below. On the other hand, it
requires the exponent γ ≥ 1/2 and does not capture the fact that even smaller moments can
be estimated for larger values of d. This motivates the inequality (1.4) with different choices
of a. As a consequence of our result, the smallest value of γ such that (1.4) holds for some
a ≥ 0 (indeed, for a = d − 1) is
2−d
1 ≤ d < 2 , γmin = 0 d > 2 .
(1.5)
2
We emphasize that we establish the inequality in these endpoint cases and that the resulting
inequality for 1 ≤ d < 2 is order-sharp in the weak coupling limit, see Remark 2.11. As one
may expect by analogy with the Euclidean situation, the case d = 2 is somewhat special,
since the minimal value of γ is 0, but the inequality is not valid in the endpoint case.
We consider also the case of a homogeneous tree, i.e., a tree where all edges have equal
length and all vertices are of the same degree. In this case, the function g0 grows exponentially and the Laplacian −∆N is positive definite. We prove Cwikel-Lieb-Rozenblum
inequalities for the number of eigenvalues that a potential V generates below the bottom of
the spectrum of −∆N .
An important ingredient in our proof of eigenvalue estimates are one-dimensional Sobolev
inequalities with weights. In particular, if the integral (1.3) is finite, we combine them with
a Sturm oscillation argument in order to deduce Cwikel-Lieb-Rozenblum inequalities. This
yields remarkably good bounds on the constants. We believe that our technique, in particular
the duality argument in Proposition 7.2, has applications beyond the context of this paper.
As we have pointed out, one of the main motivations for this work is to understand how
the dimensionality of the underlying space is reflected in eigenvalue estimates. Several results
γmin =
EIGENVALUE ESTIMATES — October 9, 2007
3
in the literature can be viewed in this light. If the global dimension of the underlying space
is, in contrast to our situation, smaller than the local dimension, then the eigenvalues are
typically estimated by a sum of two terms. Lieb-Thirring inequalities of this form have
been proved by Lieb, Solovej and Yngvason [25] for the Pauli operator. The second, nonstandard term there corresponds to states in the lowest Landau level, which are localized in
the plane orthogonal to the magnetic field. A two-term inequality of more obvious geometric
nature was proved by Exner and Weidl [12] for Schrödinger operators in a waveguide ω × R,
ω ⊂ Rd−1 . Here the second term corresponds to the global dimension, which is one, as
opposed to the local dimension d. These two-term estimates are order-sharp both in the weak
coupling regime (where the global dimension is dominant) and in the strong coupling regime
(where the local dimension is dominant). In our situation, however, the global dimension is
larger than the local dimension, and a two-term inequality would neither in the weak nor
in the strong coupling regime be order-sharp. Therefore we propose families of inequalities,
which are sharp in different coupling regimes. This is somewhat reminiscent of the family of
inequalities proved by Hundertmark and Simon [17] for the discrete Laplacian on the lattice
Zd , where the local dimension is 0 and the global dimension is d.
Acknowledgements. The authors are grateful to Robert Seiringer and Timo Weidl for
several useful discussions, and to the organizers of the workshop ‘Analysis on Graphs’ at
the Isaac Newton Institute in Cambridge for their kind invitation. This work has been
supported by FCT grant SFRH/BPD/ 23820/2005 (T.E.) and DAAD grant D/06/49117
(R.F.). Partial support by the ESF programme SPECT (T.E. and H.K.) and the DAADSTINT PPP programme (R.F.) is gratefully acknowledged.
2. Main results and discussions
2.1. Preliminaries. Let Γ be a rooted metric tree with root o. By |x| we denote the unique
distance between a point x ∈ Γ and the root o. Throughout we assume that Γ is of infinite
height, i.e., supx∈Γ |x| = ∞. The branching number b(x) of a vertex x is defined as the
number of edges emanating from x. We assume the natural conditions that b(x) > 1 for any
vertex x 6= o and that b(o) = 1.
We define the Neumann Laplacian −∆N as the self-adjoint operator in L2 (Γ) associated
with the closed quadratic form
Z
|ϕ′ (x)|2 dx, ϕ ∈ H 1 (Γ).
(2.1)
Γ
Here
and
H 1 (Γ)
consists of all continuous functions ϕ such that ϕ ∈ H 1 (e) on each edge e of Γ
Z
Γ
|ϕ′ (x)|2 + |ϕ(x)|2 dx < ∞.
The operator domain of −∆N consists of all continuous functions ϕ such that ϕ′ (o) = 0,
ϕ ∈ H 2 (e) for each edge e of Γ and such that at each vertex x =
6 o of Γ the matching
conditions
ϕ− (x) = ϕ1 (x) = · · · = ϕb(x) (x) ,
ϕ′− (x) = ϕ′1 (x) + · · · + ϕ′b(x) (x)
are satisfied. Here ϕ− denotes the restriction of ϕ on the edge terminating in x and ϕj , j =
1, . . . , b(x), denote the restrictions of ϕ to the edges emanating from x, see, e.g., [28, 27] for
details.
In this paper we are interested in Schrödinger operators −∆N − V in L2 (Γ). Throughout
we assume that the potential V is a real-valued, sufficiently regular function on Γ, the
positive part of which vanishes at infinity in a suitable sense. (We shall be more precise
4
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
below.) In this case the negative spectrum of −∆N − V consists of discrete eigenvalues of
finite multiplicities. Our goal is to estimate the total number of these eigenvalues or, more
generally, moments of these eigenvalues in terms of integrals of the potential V .
The starting point of our analysis is
Theorem 2.1. Let γ ≥ 1/2. Then there exists a constant Lγ such that for any rooted metric
tree Γ and any V ,
Z
γ+ 1
V (x)+ 2 dx.
tr(−∆N − V )γ− ≤ Lγ
(2.2)
Γ
We emphasize that the constant Lγ is independent of Γ. This result is clearly analogous
to the standard one-dimensional Lieb-Thirring inequalities. An advantage is its universality.
Moreover, we will see in Subsection 2.3 below, that the right hand side has the correct order
of growth in the strong coupling limit when V is replaced by αV and α → ∞. On the other
hand, it does not reflect the geometry of Γ at all and it does not display the correct behavior
in the weak coupling limit when V is replaced by αV and α → 0.
The main goal of this paper is to obtain eigenvalue estimates which take the global structure of Γ into account. We shall consider trees which possess certain additional symmetry
properties. Namely, we impose
Assumption 2.2. The tree Γ is regular, i.e., all the vertices at the same distance from the
root have equal branching numbers and all the edges emanating from these vertices have
equal length.
Let x be a vertex such that there are k + 1 vertices on the (unique) path between o and x
including the endpoints. We denote by tk the distance |x| and by bk the branching number
of x. Moreover, we put t0 := 0 and b0 := 1. Note that tk and bk are only well-defined for
regular trees and that these numbers, in the regular case, uniquely determine the tree.
We define the (first) branching function g0 : R+ → N by
g0 (t) := b0 b1 · · · bk ,
if tk < t ≤ tk+1 ,
k ∈ N0 .
Here N = {1, 2, 3, . . .} and N0 := N ∪ {0}. Note that g0 is a non-decreasing function and
that g0 (t) coincides with the number of points x ∈ Γ such that |x| = t. The rate of growth
of g0 reflects the rate of growth of the tree Γ. More precisely, g0 measures how the surface
of the ‘ball’ {x ∈ Γ : |x| < t} grows with t. Of great importance in our analysis will be the
fact whether the reduced height of Γ,
Z ∞
dt
(2.3)
ℓΓ :=
g0 (t)
0
is finite or not.
In addition to Assumption 2.2 we shall impose
Assumption 2.3. The function V is symmetric, i.e., for any x ∈ Γ the value V (x) depends
only on the distance |x| between x and the root o.
With slight abuse of notation we shall write sometimes V instead of V (| · |).
2.2. Eigenvalue estimates on trees. In this subsection we present our main results. We
denote by N (T ) the number of negative eigenvalues (counting multiplicities) of a self-adjoint,
lower bounded operator T . We begin with the case where the reduced height (2.3) is finite.
In this case we shall prove
EIGENVALUE ESTIMATES — October 9, 2007
5
Theorem 2.4 (CLR bounds for trees of finite reduced height). Let Γ be a regular
metric tree with ℓΓ < ∞ and let w : R+ → R+ be a positive function such that for some
2<q≤∞
Z t
2/q Z ∞
q
q−2
ds
−
g0 (s) 2 w(s) 2 ds
M := sup
<∞ .
(2.4)
g0 (s)
t≥0
0
t
Let p := q/(q − 2). Then there exists a constant Np (Γ, w) such that
Z
N (−∆N − V ) ≤ Np (Γ, w) V (|x|)p+ w(|x|) dx
(2.5)
Γ
for all symmetric V . Moreover, the sharp constant in (2.5) satisfies
1 p p
′ p−1
Np (Γ, w) ≤ (1 + p )
1+ ′ M .
p
By definition, if q = ∞ condition (2.4) is understood as
Z ∞
ds
g0 (s)
<∞,
sup sup
w(s)
g
0 (s)
t≥0 0≤s≤t
t
and one has N1 (Γ, w) ≤ M .
In order to give more explicit estimates we assume that the growth of the branching
function is sufficiently regular in the sense of
Definition 2.5. A regular metric tree Γ has global dimension d ≥ 1 if its branching function
satisfies
g0 (t)
g0 (t)
0 < c1 := inf
≤ sup
=: c2 < ∞ .
(2.6)
d−1
d−1
t≥0 (1 + t)
t≥0 (1 + t)
Obviously, if Γ has global dimension d, then it has finite reduced height if and only if
d > 2. In this case Theorem 2.4 implies
Corollary 2.6. Assume that Γ has global dimension d > 2. Then for any a ≥ 1 there exists
a constant C(a, Γ) such that for any symmetric V
Z
a
1+a
N (−∆N − V ) ≤ C(a, Γ) V (|x|) 2 g0 (|x|) d−1 dx .
Γ
Next we turn to the case of infinite reduced height ℓΓ = ∞. It is easy to see that
Schrödinger operators −∆N − V on such trees with non-trivial V ≥ 0 have at least one
negative eigenvalue, no matter how small V is. Hence it is impossible to estimate the
number of eigenvalues from above by a weighted integral norm of the potential. However,
under the assumption that the tree has a global dimension we can prove estimates for the
moments of negative eigenvalues of −∆N − V . Moreover, we can treat the case 0 ≤ a < 1
which was left open in Corollary 2.6. Our result is
Theorem 2.7 (LT bounds for trees). Let Γ be a regular metric tree with global dimension
d ≥ 1.
(1) Assume that either 1 ≤ d < 2 and 0 ≤ a ≤ d − 1, or else that d ≥ 2 and 0 ≤ a < 1.
Then for any γ ≥ 1−a
2 there exists a constant C(γ, a, Γ) such that for any symmetric
V
Z
tr(−∆N − V )γ− ≤ C(γ, a, Γ)
γ+ 1+a
2
Γ
V (|x|)+
a
g0 (|x|) d−1 dx.
(2.7)
(2) Assume that either 1 ≤ d < 2 and a > d − 1, or else that d = 2 and a ≥ 1. Then for
any γ > (1 + a) 2−d
2d there exists C(γ, a, Γ) such that (2.7) holds for any symmetric
V.
6
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
(3) Assume that d > 2 and that a ≥ 1. Then for any γ ≥ 0 there exists C(γ, a, Γ) such
that (2.7) holds for any symmetric V .
One can prove that our conditions on γ are not only sufficient but (except for the limiting
case in Part (2)) also necessary for the validity of (2.7). This is further discussed in Subsection 2.3. Part (3) is in fact an immediate consequence of Corollary 2.6 and an argument by
Aizenman and Lieb [2]. It is stated here for the sake of completeness.
If the branching function g0 grows ‘very’ fast, the Laplacian −∆N is positive definite. In
this case it is reasonable not only to estimate the number of negative eigenvalues of −∆N −V ,
but also the number of eigenvalues less then the bottom of the spectrum of −∆N . We carry
through this analysis for a special class of trees.
A regular metric tree is called homogeneous if all the edges have the same length τ and
if the branching number bk = b > 1 is independent of k. Homogeneous trees correspond
intuitively to trees of infinitely large global dimension. By scaling it is no loss of generality
to assume that τ = 1. The branching function g0 then reads
g0 (t) = bj ,
j < t ≤ j + 1,
j ∈ N0 .
The Laplacian −∆N (or rather its Dirichlet version) on a homogeneous tree was studied
in [32]. It follows from the analysis there that −∆N is positive definite and its essential
spectrum starts at
λb =
1
arccos
Rb
2
1
1
,
Rb =
b 2 + b− 2
.
2
We shall prove
Theorem 2.8 (CLR bounds for homogeneous trees). Let Γ be a homogeneous tree
with edge length 1 and branching number b > 1 and let w : R+ → R+ be a positive function
such that for some 2 < q ≤ ∞
−1
M := sup (1 + t)
t≥0
Z
t
q
(1 + s) w
0
− q−2
2
ds
2/q
.
Let p = q/(q − 2). Then there exists a constant Np (b, w) such that
Z
N (−∆N − V − λb ) ≤ Np (b, w) V (|x|)p+ w(|x|) dx
(2.8)
Γ
for all symmetric V . Moreover, the sharp constant in (2.8) satisfies
1 p p
′ p−1
Np (b, w) ≤ C(b) (1 + p )
1+ ′ M
p
(2.9)
with some constant C(b) depending only on b.
Choosing w(t) = (1 + t)a we obtain the following strengthening of Corollary 2.6.
Corollary 2.9. Let Γ be a homogeneous tree with edge length 1 and branching number b > 1.
Then for any a ≥ 1 there exists a constant C(a, b) such that for any symmetric V
Z
1+a
N (−∆N − V − λb ) ≤ C(a, b) V (|x|)+2 (1 + |x|)a dx .
Γ
EIGENVALUE ESTIMATES — October 9, 2007
7
2.3. Discussion. In this subsection we discuss the inequality (2.7) and the conditions for
its validity given in Theorem 2.7.
Remark 2.10 (Strong coupling limit). Inequality (2.7) with a = 0 coincides with (2.2),
Z
1
γ+ 1
γ
V (|x|)+ 2 dx, γ ≥ .
tr(−∆N − V )− ≤ Lγ
2
Γ
This inequality reflects the correct behavior in the strong coupling limit. Indeed, if V is, say,
continuous and of compact support then standard Dirichlet-Neumann bracketing [31, Thm.
XIII.80] leads to the Weyl-type asymptotic formula
Z
1
γ+ 1
(2.10)
V (|x|)+ 2 dx, γ ≥ 0,
lim α−γ− 2 tr (−∆N − αV )γ− = Lcl
γ,1
α→∞
Γ
with
Γ(γ + 1)
Lcl
.
γ,1 := √
2 π Γ(γ + 3/2)
This shows in particular that (2.7) can not hold for a < 0.
(2.11)
Remark 2.11 (Weak coupling limit). Assume that Γ has global dimension d ∈ [1, 2).
Inequality (2.7) with a = d − 1, γ = (2 − d)/2 reads
Z
2−d
2−d
2
V (|x|)+ g0 (|x|) dx.
, d − 1, Γ
tr(−∆N − V )− ≤ C
2
Γ
This inequality reflects the correct behavior in the weak coupling limit.R Indeed, it is shown
in [19] that −∆N − αV has at least one negative eigenvalue whenever Γ V (|x|) dx > 0, and
that for α sufficiently small this eigenvalue, say λ1 (α), is unique and satisfies
2
2
−a1 α 2−d ≤ λ1 (α) ≤ −a2 α 2−d ,
α → 0,
(2.12)
for suitable constants a1 ≥ a2 > 0 depending on V . This fact shows also that (2.7) does not
hold for 1 ≤ d < 2, a ≥ 0 and γ < (1 + a) 2−d
2d . We do not know whether (2.7) holds in the
2−d
endpoint case γ = (1 + a) 2d when 1 ≤ d < 2 and a > d − 1.
Similarly, when Γ has global dimension
d = 2, one can show that −∆N − αV has at least
R
one negative eigenvalue whenever Γ V (|x|) dx > 0. Hence (2.7) does not hold for d = 2,
a ≥ 0 and γ = 0.
Remark 2.12 (Dirac-potential limit). As we have seen in the previous remark, the condition γ > (1 + a)(2 − d)/(2d) in Part (2) of Theorem 2.7 comes from the weak coupling
limit. Now we explain that the condition γ ≥ (1 − a)/2 in Part (1) comes from what may
be called the Dirac-potential limit. Consider the sequence of potentials Vn = nχ(0,n−1 ) .
Using a trial function supported near the root o one easily proves that tr(−∆N − Vn )γ− is
R γ+ a+1 a
bounded away from zero uniformly in n. On the other hand, Vn 2 g0(d−1) dx tends to
zero if γ < (1 − a)/2. This shows that the condition γ ≥ 1−a
2 is necessary for the validity of
(2.7).
Remark 2.13 (Slowly decaying potentials). Assume that V is a symmetric function which
is locally sufficiently regular and obtains the asymptotics V (t) ∼ αt−s as t → ∞ for some
s > 0, α > 0. By standard methods (see, e.g., [31, Thm. XIII.6]) one shows that the operator
−∆N −V has only a finite number of negative eigenvalues provided s > 2. However, the semiclassical expression for the number of negative eigenvalues, i.e. the right hand side of (2.10)
with γ = 0, is only finite under the more restrictive condition s > 2d. Our Corollary 2.6
with sufficiently large a gives a quantitative estimate on the number of negative eigenvalues
for the whole range of exponents s > 2 if d > 2. Similarly, in the case 1 ≤ d ≤ 2 we
8
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
obtain quantitative information about the magnitude of the eigenvalues, which goes beyond
semi-classics.
Remark 2.14 (Dirichlet boundary conditions). The reader might wonder how our main
theorems change, if a Dirichlet instead of a Neumann boundary condition is imposed at the
root. Let −∆D be the self-adjoint operator in L2 (Γ) generated by the quadratic form (2.1)
with form domain H01 (Γ) := {φ ∈ H 1 (Γ) : φ(0) = 0}. By the variational principle, any
bound for −∆N − V implies a bound for −∆D − V . However, it turns out that inequalities
for the latter operator hold for a strictly larger range of parameters. Indeed, the analog of
Theorems 2.4 states that the inequality
Z
a
γ+ 1+a
γ
tr(−∆D − V )− ≤ C(γ, a, Γ) V (|x|)+ 2 g0 (|x|) d−1 dx.
Γ
holds provided either 0 ≤ a < 1 and γ ≥ (1 − a)/2, or else a ≥ 1 and γ ≥ 0 and d 6= 2, or
else a ≥ 1 and γ > 0 and d = 2. This follows (except for the statement for γ = 0, 1 ≤ d < 2)
from Theorem 7.4. There is also an analog of Theorem 2.4 for −∆D which is obtained by
simply interchanging the two intervals of integration in the assumption (2.4). We omit the
details. For spectral asymptotics of the operator −∆D − V we refer to [27].
2.4. One-dimensional Schrödinger operators with metric. Our symmetry assumptions will allow us to reduce the spectral analysis of the operator −∆N − V to the spectral
analysis of a family of one-dimensional Schrödinger-type operators. The main ingredient in
the proof of Theorem 2.7 will be an inequality for such operators, which is of independent
interest.
We consider a positive, measurable and locally bounded function g on [0, ∞) and denote
1 (R ) such that
by H 1 (R+ , g) the space of all functions f ∈ Hloc
+
Z ∞
|f ′ (t)|2 + |f (t)|2 g(t) dt < ∞.
0
The quadratic form
Z
0
∞
|f ′ (t)|2 g(t) dt
(2.13)
with form domain H 1 (R+ , g) defines a self-adjoint operator Ag in L2 (R+ , g). Note that this
operator corresponds to the differential expression
d d
g ,
dt dt
and that functions f in its domain satisfy Neumann boundary conditions f ′ (0) = 0 at the
origin (at least when g is sufficiently regular near 0).
For our first results we assume that g grows sufficiently fast in the sense that
Z ∞
ds
<∞
∀ t > 0.
(2.14)
g(s)
t
Ag = −g−1
We shall prove that under this condition the number of negative eigenvalues of the Schrödinger operators Ag − V can be estimated in terms of weighted Lp -norms of V . More precisely,
one has
Theorem 2.15. Assume (2.14) and let w : R+ → R+ be a positive function such that for
some 2 < q ≤ ∞
2/q Z ∞
Z t
q−2
q
ds
−
<∞ .
(2.15)
g(s) 2 w(s) 2 ds
M := sup
g(s)
t≥0
t
0
EIGENVALUE ESTIMATES — October 9, 2007
Let p := q/(q − 2). Then the inequality
N (Ag − V ) ≤ Cp (w, g)
Z
∞
0
V+p w dt
9
(2.16)
holds for all V , and the sharp constant Cp (w, g) in (2.16) satisfies
1 p p
p
′ p−1
M ≤ Cp (w, g) ≤ 1 + p
1+ ′ M .
p
Moreover, if M = ∞ then there is no constant Cp (w, g) such that (2.16) holds for all V .
By definition, if q = ∞ condition (2.15) is understood as
Z ∞
g(s)
ds
M := sup sup
< ∞,
w(s)
g(s)
t≥0 0≤s≤t
t
and the sharp constant is C1 (w, g) = M . This leads to the following beautiful estimate.
R∞
Example 2.16. Taking w(t) = g(t) t g−1 (s) ds and q = ∞ one obtains
Z ∞
Z ∞
ds
dt ,
(2.17)
V (t)+ g(t)
N (Ag − V ) ≤
g(s)
t
0
which is sharp (meaning that the estimate is no longer true for all g and all V if the right
hand side is multiplied by a constant less than one). As a consequence one also finds
Z ∞ Z ∞
dt
N (Ag − V ) ≤
V+ g dt .
g 0
0
Theorem 2.15 gives a complete characterization of weights for which the number of negative eigenvalues can be estimated by a weighted norm of the potential. When g grows very
fast, the operator Ag will be positive definite and in this case one may not only ask for the
number of eigenvalues of Ag − V below 0 but also below the bottom of the spectrum of Ag .
We turn to this question next. We assume, in addition to (2.14), that
Z ∞
Z t
ds
< ∞.
(2.18)
sup
g(s) ds
g(s)
t>0 0
t
This condition is necessary and sufficient for the operator Ag to be positive definite, see
Proposition 5.1 below or [33, Thm. 5.2]. We denote the bottom of its spectrum by λ(Ag ) > 0
and assume that λ(Ag ) is not an eigenvalue of Ag . Let ω be the unique (up to a constant)
distributional solution of the differential equation
−(gω ′ )′ = λ(Ag ) g ω
on R+
(2.19)
satisfying the boundary condition ω ′ (0) = 0. Since λ(Ag ) is not an eigenvalue, the function
ω is not square-integrable with respect to the weight g. We quantify the growth of ω 2 g by
assuming that
Z
∞
0
ω −2 g−1 ds < ∞.
(2.20)
Under these conditions one has
Theorem 2.17. Assume (2.14), (2.18) and (2.20). Let w : R+ → R+ be a positive function
such that for some 2 < q ≤ ∞
2/q Z ∞
Z t
q q2 − q−2
2
ω −2 g−1 ds < ∞,
ds
ω g w
M := sup
t>0
and put p :=
q
q−2 .
0
t
Then the inequality
N (Ag − V − λ(Ag )) ≤ Cp (w, g, ω)
Z
∞
0
V+p w dt
(2.21)
10
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
holds for all V , and the sharp constant Cp (w, g, ω) satisfies
1 p p
p
′ p−1
M ≤ Cp (w, g, ω) ≤ 1 + p
1+ ′ M .
p
(2.22)
Finally, we present some estimates without imposing the condition (2.14). It is easy to
see that if the integral in (2.14) is infinite, then Ag − V will have a negative eigenvalue for
any non-negative V 6≡ 0, hence no estimate on the number of eigenvalues in terms of norms
of V can hold. Below we shall prove that estimates on moments of eigenvalues do hold. For
the sake of simplicity we restrict ourselves to the case where g has power-like growth, i.e.,
g(t)
g(t)
≤ sup
=: c2 < ∞
d−1
d−1
t>0 (1 + t)
t>0 (1 + t)
0 < c1 := inf
(2.23)
for some d ≥ 1. Note that (2.14) holds iff d > 2. We shall consider inequalities of the form
Z ∞
γ+ a+1
γ
V (t)+ 2 (1 + t)a dt,
tr(Ag − V )− ≤ L
L = L(γ, a, d, c1 , c2 ).
(2.24)
0
In Remark 7.3 below we show that the relation between the exponent of V and that of the
weight (1 + t) can not be improved. Our result is
Theorem 2.18. Assume (2.23) for some d ≥ 1.
(1) Let either 1 ≤ d < 2 and 0 ≤ a ≤ d − 1, or else d ≥ 2 and 0 ≤ a < 1. Then (2.24)
holds iff γ ≥ (1 + a)/2.
(2) Let either 1 ≤ d < 2 and a > d − 1, or else d = 2 and a ≥ 1. Then (2.24) holds iff
γ > (1 + a)(2 − d)/(2d).
(3) Let d > 2 and a ≥ 1. Then (2.24) holds for any γ ≥ 0.
Part (3) is of course a consequence of Theorem 2.15 (for γ = 0) and of an argument
by Aizenman and Lieb [2] (for γ > 0). Note carefully that for small a (Part (1)) the
inequality (2.24) holds in the endpoint case, while it does not for large a (Part (2)). This is
a phenomenon due to the Neumann boundary conditions which is not present when Dirichlet
boundary conditions are imposed instead, see Theorem 7.4.
2.5. Outline of the paper. This paper is organized as follows. In Section 3 we prove
Theorem 2.1 and a weighted version of it about arbitrary, not necessarily regular, metric
trees. In Section 4 we show how our main results, Theorems 2.4, 2.7 and 2.8, follow from
the results about one-dimensional Schrödinger operators in Subsection 2.4. In Section 5 we
give the proofs of Theorems 2.15 and 2.17. Section 6 is of auxiliary character and contains
the proof of a family of Sobolev interpolation inequalities which will be useful in the proof
of Theorem 2.18. Finally, in Section 7 we will use a duality argument and estimates for
Dirichlet eigenvalues in order to obtain the statements of Theorem 2.18.
3. Eigenvalue estimates on general metric trees
This section is devoted to the proof of Theorem 2.1. Moreover, we shall also prove the
following weighted analog.
Theorem 3.1. Let a > 0 and γ > (1 + a)/2. Then there exists a constant Ca (γ) such that
Z
γ+ 1+a
γ
(3.1)
V (x)+ 2 |x|a dx.
tr(−∆N − V )− ≤ Ca (γ)
Γ
We emphasize that the constant in (3.1) can be chosen independently of the tree. For the
proofs of Theorems 2.1 and 3.1 we use the following results about half-line operators.
EIGENVALUE ESTIMATES — October 9, 2007
11
Proposition 3.2. Let Γ = R+ and a ≥ 0. Let γ > (1 + a)/2 if a > 0 and γ ≥ 1/2 if a = 0.
Then there exists a constant LEK
γ,a such that
tr (−∆N − V
)γ−
≤
LEK
γ,a
Z
∞
0
γ+ 1+a
a
2
V (t)+
t dt
(3.2)
for all V .
To prove (3.2) we extend V to an even function W on R. Then the left hand side of
(3.2) can be estimated from above by the corresponding moments of the whole-line operator
−d2 /dx2 − W , and the claimed inequality for that operator follows from [7] and [34]. Using
in addition the sharp constants from [16] and [2] one obtains for a = 0 the following bounds
on the constants,
1
cl
LEK
γ,0 ≤ 4 Lγ,1 if γ ≥ ,
2
cl
LEK
γ,0 ≤ 2 Lγ,1 if γ ≥
3
2
(3.3)
with Lcl
γ,1 from (2.11). Note that the inequality (3.2) with this constant for γ = 1/2 and
a = 0 is sharp, and therefore so is (2.2) for γ = 1/2. Now we turn to the
Proof of Theorems 2.1 and 3.1. The idea is to impose Neumann boundary condition at all
but one emanating edges of all vertices. This decreases the operator −∆N −V . The resulting
operator can be identified with a direct sum of half-line operators for which one can use
Proposition 3.2.
S
To be more precise, we decompose the graph Γ = j Γj into a disjoint union of infiL
P 1
1
nite halflines Γj . Then L2 (Γ) =
j L2 (Γj ) and H (Γ) ⊂
j H (Γj ). By the variational
principle, this implies
M
Γ
−∆N − V ≥
−∆Nj − Vj ,
j
Γ
where −∆Nj is the Neumann Laplacian on Γj and Vj is the restriction of V to Γj . Hence
Proposition 3.2 yields
γ
X
Γ
tr(−∆N − V )γ− ≤
trL2 (Γj ) −∆Nj − Vj
−
j
≤ LEK
γ,α
≤ LEK
γ,α
XZ
j
Z
Γ
Γj
γ+ 1+a
2
Vj (x)+
γ+ 1+a
2
V (x)+
dist(x, ∂Γj )a dx
|x|a dx,
as claimed.
4. Eigenvalue estimates on regular trees
In this section we show how our main results, Theorems 2.4, 2.7 and 2.8, can be deduced
from the results about one-dimensional Schrödinger operators in Subsection 2.4. To do so, we
exploit the symmetry of the tree and the potential, which allows us to decompose −∆N − V
into a direct sum of half-line Schrödinger operators in weighted L2 -spaces. We recall this
construction next.
12
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
4.1. Orthogonal decomposition. In this subsection we recall the results of Carlson [5]
and of Naimark and Solomyak [27, 28]. We need some notation. For each k ∈ N we define
the higher order branching functions gk : R+ → N0 by

t < tk ,
 0,
gk (t) :=
1,
tk ≤ t < tk+1 ,

bk+1 bk+2 · · · bn , tn ≤ t < tn+1 , k < n ,
and introduce the weighted Sobolev space H01 ((tk , ∞), gk ) as the closure of C0∞ (tk , ∞) in the
norm
1
Z ∞
2
′
2
2
|f (t)| + |f (t)| gk (t) dt .
tk
Let Ak be the self-adjoint operator in L2 ((tk , ∞), gk ) given by the quadratic form
Z ∞
|f ′ (t)|2 gk (t) dt
ak [f ] :=
tk
with form domain H01 ((tk , ∞), gk ). Notice that the operators Ak with k ≥ 1 satisfy Dirichlet
boundary condition at tk , while the operator A0 satisfies Neumann boundary condition at
t0 = 0.
The following statement is taken from [28] and [33].
Proposition 4.1. Let V ∈ L∞ (Γ) be symmetric. Then −∆N − V is unitarily equivalent to
the orthogonal sum of operators
−∆N − V ≃ (A0 − V ) ⊕
∞
X
k=1
⊕ A k − Vk
[b1 ...bk−1 (bk −1)]
.
(4.1)
Here the symbol [b1 ...bk−1 (bk − 1)] means that the operator Ak − Vk appears b1 ...bk−1 (bk − 1)
times in the orthogonal sum, and Vk denotes the restriction of V to the interval (tk , ∞).
4.2. Proof of Theorems 2.4 and 2.7. Let us compare the operators Ak with each other.
From the definition of the function gk it follows that
R∞
R∞
′ 2
2
′ 2
2
tk |f | − Vk |f | gk dt
tk |f | − Vk |f | g0 dt
R∞
R∞
=
.
2
2
tk |f | gk dt
tk |f | g0 dt
Since every function f ∈ H01 ((tk , ∞), gk ) can be extended by zero to a function in H 1 (R+ , g0 ),
the variational principle shows that
tr(Ak − Vk )γ− ≤ tr(A0 − χ(tk ,∞)V )γ−
(4.2)
for any k ∈ N and γ ≥ 0.
Assuming the validity of Theorems 2.15 and 2.18 we now give the
Proof of Theorems 2.4 and 2.7. In the case of Theorem 2.4 put γ = 0 and let q and w be
such that (2.4) holds. Moreover, put p = q/(q − 2). In the case of Theorem 2.7 let γ be as
indicated there and put p = γ + (1 + a)/2 and w(t) := g0 (t)a/(d−1) . It follows from Theorems
2.15 and 2.18, respectively, that in both cases there exists a constant C such that
Z ∞
V (t)p+ w(t) dt
tr(A0 − V )γ− ≤ C
0
EIGENVALUE ESTIMATES — October 9, 2007
13
for all V . Combining this with the orthogonal decomposition (4.1) and inequality (4.2) we
obtain
tr(−∆N − V
)γ−
= tr(A0 − V
≤C
Z
∞
0
+C
∞ X
∞ Z
X
k=0
=C
Z
Γ
+
∞
X
k=1
b1 · · · bk−1 (bk − 1) tr(Ak − χ(tk ,∞)V )γ−
V (t)p+ w(t) dt
k=1
=C
)γ−
b1 · · · bk−1 (bk − 1)
tk+1
tk
Z
∞
V
tk
(t)p+ w(t) dt
(b0 · · · bk )V (t)p+ w(t) dt
V (|x|)p+ w(|x|) dx,
as claimed.
4.3. Proof of Theorem 2.8. In this subsection we assume that g0 is the first branching
function of a homogeneous metric tree with edge length 1 and branching number b > 1.
Denote by λb the bottom of its essential spectrum and by ω the function on R+ satisfying
in distributional sense
−(g0 ω ′ )′ = λb g0 ω ,
ω ′ (0) = 0,
ω(j+) = ω(j−),
ω ′ (j−) = bω ′ (j+),
j ∈ N.
In the proof of Theorem 2.8 we need the following technical result.
Lemma 4.2. There exist constants 0 < C1 < C2 < ∞ such that
1+t
1+t
C1 p
≤ ω(t) ≤ C2 p
,
g0 (t)
g0 (t)
t ≥ 0.
(4.3)
Assuming this for the moment we give the
Proof of Theorem 2.8. Proceeding in the same way as in the proof of Theorems 2.4 and 2.7
one sees that it suffices to prove that
Z ∞
(4.4)
V (t)p+ w(t) dt .
N (A0 − V − λb ) ≤ C
0
We shall deduce this from Theorem 2.17 with g = g0 . By the explicit form of g0 we see
that (2.14) and (2.18) are satisfied. Moreover, λb = λ(A0 ) and ω is the generalized ground
state of A0 in the sense of (2.19). It follows from Lemma 4.2 that the assumption (2.20) is
satisfied and that one has
2/q Z ∞
Z t
q
q 2 − q−2
ω −2 g−1 ds
ω g0 w 2 ds
t
0
≤
C2
C1
2 Z
Hence (4.4) follows from Theorem 2.17.
We are left with the
t
q
(1 + s) w
0
− q−2
2
ds
2/q
1
.
1+t
14
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
Proof of Lemma 4.2. A direct calculation shows that
with µ :=
√
ω(t) = αj cos(µ(t − j)) + βj cos(µ(j + 1 − t)),
j < t < j + 1,
λb , α0 := 1, β0 := 0 and
−αj−1 = b βj .
αj−1 cos µ + βj−1 = αj + βj cos µ ,
This can be rewritten as
αj
βj
b
− 21
2
1
−b− 2
and by induction one easily finds that
j
αj
j+1
b− 2
1
βj
−j b− 2
1
b2
0
!
1
j b2
−j + 1
αj−1
βj−1
!
α0
β0
,
.
This implies
− 12
ω(t) = g0 (t)
if j < t < j + 1, and hence
j −1
b 2 cos(µ(j + 1 − t))
(j + 1) cos(µ(t − j)) −
j+1
1
ω(t) ∼ g0 (t)− 2 (1 + t) ϕ(t),
t → ∞,
(4.5)
where ϕ is periodic with period 1 and
1
ϕ(t) = cos µt − b− 2 cos(µ(1 − t)),
The estimates
1
1
1
b 2 − b− 2
− 21
0 < t < 1.
1
b 2 − b− 2
0 < t < 1,
1
1
1 ≥ ϕ(t) ≥ b
1 > 0,
b 2 + b− 2
b 2 + b− 2
and the asymptotics (4.5) imply that (4.3) holds for all sufficiently large t. On the other
hand, by the Sturm oscillation theorem (or by direct calculation) ω is bounded and bounded
away from zero on compacts. This proves the lemma.
5. Estimates on the number of eigenvalues
5.1. Proof of Theorem 2.15. Our goal in this section is to prove the statements of Theorem 2.15. An important ingredient will be weighted Hardy-Sobolev inequalities. The
characterization of all admissible weights is independently due to Bradley, Maz’ya and Kokilashvili. The constant in (5.3) below is due to Opic. We refer to [29, Thm. 6.2] for the
proof and further historical remarks.
Proposition 5.1. Let 2 ≤ q ≤ ∞. The inequality
2/q
Z ∞
Z
2
q
≤S
|w(r)u(r)| dr
0
0
∞
|v(r)u′ (r)|2 dr
(5.1)
holds for all absolutely continuous functions u on [0, ∞) with limr→∞ u(r) = 0 if and only if
1/2
1/q Z ∞
Z r
−2
q
< ∞.
(5.2)
|v(s)| ds
|w(s)| ds
T := sup
r>0
0
r
In this case, the sharp constant S in (5.1) satisfies
2 1/2
q 1/q
T.
1+
T ≤S ≤ 1+
2
q
(5.3)
EIGENVALUE ESTIMATES — October 9, 2007
If q = ∞, then (5.2) means
Z
T := sup sup |w(s)|
r>0
0≤s≤r
∞
r
−2
|v(s)|
ds
1/2
15
< ∞,
and in (5.3) one has T = S. Now everything is in place to give the
Proof of Theorem 2.15. Let w ≥ 0 such that M defined in (2.15) is finite. Then Proposition
5.1 yields for all u ∈ H 1 (R+ , g),
Z ∞
2/q
Z ∞
q q2 − q−2
2
2
|u| g w
≤S
|u′ |2 g dt,
(5.4)
dt
0
0
where
q 2/q
2
M ≤S ≤ 1+
1+
M.
2
q
We now use an argument in the spirit of [14] to deduce (2.16) from (5.4). Let ω be the solution
of −(gω ′ )′ − V ωg = 0 that satisfies the boundary condition ω ′ (0) = 0. By Sturm-Liouville
theory (see, e.g., [35, Thm. 14.2]) the number of zeros of ω coincides with the number N of
negative eigenvalues of Ag − V . Denote these zeros by 0 < a1 < a2 < . . . < aN < ∞ and
apply (5.4) to uωχ(aj ,aj+1 ) . Integrating by parts and using Hölder’s inequality (noting that
1/p + 2/q = 1) we obtain
!2/q
Z aj+1
Z aj+1
Z aj+1
q−2
q
2
′ 2
2
q 2 − 2
dt
≤S
|ω | g dt = S
V |ω|2 g dt
|ω| g w
2
aj
aj
≤S
Z
2
aj+1
p
V w dt
aj
This implies that
1 ≤ S 2p
Z
!1/p
aj+1
Z
aj+1
aj
aj
q
|ω|q g 2 w−
V p w dt ,
q−2
2
dt
!2/q
.
∀ j = 1, . . . N .
aj
Summing this inequality over all intervals (aj , aj+1 ) we obtain
Z ∞
2p
N (Ag − V ) ≤ S
V+p w dt.
0
This proves (2.16) and shows that the sharp constant satisfies C(w) ≤ S 2p . The lower bound
C(w) ≥ S 2p follows from Theorem 7.1 below. This implies also that (2.16) does not hold if
M = ∞ and completes the proof.
For later reference we include
Example 5.2. Assume that g satisfies (2.23) for some d > 2. Then for any 1 ≤ a < ∞
Z ∞ 1+a
V+ 2 (1 + t)a dt
N (Ag − V ) ≤ Ca
0
where
and
c1
c2
1+a
2
1+a
2
Ma
Ma := sup
t>0
=
Z
a−1
a+1
≤ Ca ≤
t
(1 + s)
(2a)a
(a + 1)
a+1
2
(d−1)(a+1)−2a
a−1
0
(a − 1)
ds
a−1
2
a−1
Z
a+1
t
a−1
a+1
2a
(d − 2)− a+1 .
∞
c2
c1
1+a
2
1+a
Ma 2 .
(1 + s)−d+1 ds
16
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
(For a = 1 one has (c1 /c2 )M1 ≤ C1 ≤ (c2 /c1 )M1 and M1 := (d − 2)−1 .) This follows by
choosing w(t) = (1 + t)a and q = 2(a + 1)/(a − 1) after elementary calculations.
It is also illustrative to include another proof of estimate (2.17) in Example 2.16: The
Birman-Schwinger principle implies
1
1
−1
2
2
N (Ag − V ) ≤ trL2 (R+ ,gdt) V+ Ag V+ .
(5.5)
1
1
2
Since the operator V+2 A−1
g V+ is non-negative, we have
Z ∞
1
1
−1
2
2
trL2 (R+ ,gdt) V+ A V+ =
G(t, t) V (t)+ g(t) dt,
(5.6)
0
where G(t, t) is the diagonal of the Green function of the operator A. It follows from SturmLiouville theory (see, e.g., [35, Thm. 7.8]) that
G(t, t) =
u1 (t) u2 (t)
,
g(t)W (t)
where u1 , u2 are two linearly independent solutiuons of −(gu′ )′ = 0 and W = u′1 u2 − u1 u′2 is
their Wronskian. A direct calculation gives
Z ∞
ds
1
u1 (t) = 1, u2 (t) =
, W (t) =
.
g(s)
g(t)
t
In view of (5.5) and (5.6) this yields estimate (2.17).
5.2. Proof of Theorem 2.17. In this subsection we are working under the assumptions
(2.14), (2.18) and (2.20) of Theorem 2.17. Recall that ω is the ‘ground state’ of the operator
A. Since g may be non-smooth (it is a step function in the case of the tree) the differential
equation (2.19) has to be understood in quadratic form sense, i.e.,
Z ∞
Z ∞
′ ′
ωf g dt
(5.7)
ω f g dt = λ(A)
0
0
H 1 (R
for all f ∈
+ , g) with compact support in [0, ∞). The following identity is usually
called ground state representation.
Lemma 5.3. For any h = ω −1 f ∈ ω −1 H 1 (R+ , g),
Z
Z ∞
Z ∞
2
′ 2
|f | g dt =
|f | g dt − λ(A)
0
0
∞
0
|h′ |2 ω 2 g dt.
(5.8)
We include a sketch of the proof for the sake of completeness.
Proof. It suffices to consider h ∈ C0∞ (R+ ). Then
|(ωh)′ |2 = ω 2 |h′ |2 + ω ′ (ω|h|2 )′
and (5.8) follows from (5.7) with f = ω|h|2 .
With (5.8) at hand we can proceed to the
Proof of Theorem 2.17. We denote by B the operator in L2 (R+ , ω 2 g) corresponding to the
quadratic form
Z ∞
|h′ |2 ω 2 g dt
0
with form domain H 1 (R+ , ω 2 g). Then by the ground state representation (5.3) and Glazman’s lemma (see e.g. [4, Thm. 10.2.3])
N (A − V − λ(A)) = N (B − V ),
and the result follows from Theorem 2.15.
EIGENVALUE ESTIMATES — October 9, 2007
17
6. Sobolev interpolation inequalities
In this section we fix a parameter d ≥ 1 and study inequalities of the form
θ
Z
2/q
Z
′ 2
d−1
q
βq−1
|u | (1 + t)
dt
≤ K(q, β, d)
|u| (1 + t)
dt
Z
2
d−1
|u| (1 + t)
dt
(6.1)
1−θ
for all u ∈ H 1 (R+ , (1 + t)d−1 ). We are interested in the values of β and q for which this
inequality holds. We always fix
d − 2β
θ :=
.
(6.2)
2
In the endpoint case q = ∞ we use the convention that (6.1) means
1−θ
θ Z
Z
2
d−1
′ 2
d−1
2
2β
|u| (1 + t)
dt
|u | (1 + t)
dt
sup |u| (1 + t) ≤ K(∞, β, d)
for all u ∈ H 1 (R+ , (1 + t)d−1 ). Note that this makes sense even in the special case β = 0
(where the product βq in (6.1) is not well-defined).
d
d−2
2 ≤ β ≤ 2.
β ≤ d−1
2 , or if d
Theorem 6.1. Let d ≥ 1 and
d−1
(1) If 1 < d ≤ 2 and 0 <
> 2 and d−2
2 ≤ β ≤ 2 , then (6.1) holds for
all 2 ≤ q ≤ ∞.
d
d−1 −1
.
(2) If d ≥ 1 and d−1
2 < β ≤ 2 , then (6.1) holds for all 2 ≤ q ≤ β − 2
(3) If 1 ≤ d < 2 and β = 0, then (6.1) holds for q = ∞.
(4) If 1 ≤ d ≤ 2 and − 2−d
2 ≤ β ≤ 0, then (6.1) does not hold for 2 ≤ q < ∞.
2−d
(5) If 1 ≤ d < 2 and − 2 ≤ β < 0, or if d = 2 and β = 0, then (6.1) does not hold for
q = ∞.
d
d−1 −1
< q ≤ ∞.
(6) If d ≥ 1 and d−1
2 < β ≤ 2 , then (6.1) does not hold for β − 2
We refer to Figure 1 below for the region of allowed parameters.
Remark 6.2. In (6.1) the exponent βq − 1 of the weight on the left hand side is coupled
to the interpolation exponent θ in (6.2). This is in a certain sense optimal. Indeed, if the
inequality
θ
Z
2/q
Z
|u′ |2 (1 + t)d−1 dt
≤K
|u|q (1 + t)σ−1 dt
Z
2
d−1
|u| (1 + t)
dt
1−θ
holds for some σ > 0 and all u ∈ H 1 (R+ , (1 + t)d−1 ), then necessarily σ ≤ q(d − 2θ)/2.
(To see this put u(t) = v(lt) and let l → 0.) Note that with the value (6.2) of θ one has
q(d − 2θ)/2 = βq.
We break the proof into several lemmas which prove inequality (6.1) in the endpoint cases.
Lemma 6.3. If 1 < d ≤ 2 and 0 < β ≤
holds for q = 2 with the constant
d−1
2 ,
or if d > 2 and
K(2, β, d) = β −d+2β .
d−2
2
≤β ≤
d−1
2 ,
then (6.1)
18
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
Proof. Integration by parts shows
Z
Z
2
2β−1
−1
|u| (1 + t)
dt = (−β) ℜ uu′ (1 + t)2β − 1 dt
Z
≤ β −1 |u||u′ |(1 + t)2β dt.
We shall assume now that β < d−1
2 . The proof in the case of equality follows along the same
d−2β
lines. Then p := d−1−2β satisfies 1 < p < ∞, and by Hölder we can continue to estimate
1/p
Z
Z
2
2β−1
−1
2
2β−1
|u| (1 + t)
dt ≤ β
|u| (1 + t)
dt
×
Z
|u|
p−2
p−1
′
|u |
p
p−1
(1 + t)
2β(p−1)+1
p−1
p−1
p
dt
.
By the definition of p one has
(d − 1)(p − 2) (d − 1)p
2β(p − 1) + 1
=
+
,
p−1
2(p − 1)
2(p − 1)
and hence again by Hölder,
Z
2β(p−1)+1
p
p−2
|u| p−1 |u′ | p−1 (1 + t) p−1 dt
≤
Z
2
(d−1)
|u| (1 + t)
dt
p−2
2(p−1)
Z
′ 2
(d−1)
|u | (1 + t)
dt
This proves the inequality with the claimed constant.
p
2(p−1)
d−2
Lemma 6.4. If 1 < d ≤ 2 and 0 < β ≤ d−1
2 , or if d > 2 and 2 ≤ β ≤
holds for q = ∞ with the constant
d−2β 2
d − 1 − 2β d−1−2β
K(∞, β, d) =
.
d − 2β
2β
Here we use the convention that 00 = 1. Hence for β =
2
. Our assumptions imply that
Proof. Let p := d−2β
d > 2. By Schwarz we estimate
Z ∞
p
|u|p−1 |u′ | ds
|u(t)| ≤ p
2
d
d−1
2
.
d−1
2 ,
then (6.1)
one has K(∞, d−1
2 , d) = 2.
< p ≤ 2 if 1 < d ≤ 2 and 1 ≤ p ≤ 2 if
t
≤p
Z
∞
0
|u′ |2 (1 + s)d−1 ds
1/2 Z
∞
t
|u|2(p−1) (1 + s)−d+1 ds
d−2
2
1/2
This proves the assertion if p = 1, i.e., β =
and d > 2. If p = 2 the assertion follows
from the estimate
Z ∞
Z ∞
2(p−1)
−d+1
−2(d−1)
|u|
(1 + s)
ds ≤ (1 + t)
|u|2(p−1) (1 + s)d−1 ds.
t
0
In the remaining case 1 < p < 2 we use Hölder to obtain
Z ∞
|u|2(p−1) (1 + s)−d+1 ds
t
≤
=
Z
∞
−
(1 + s)
(d−1)p
2−p
ds
t
2−p
dp − 2
2−p
2−p Z
−dp+2
(1 + t)
Z
∞
0
∞
0
2
d−1
|u| (1 + s)
2
d−1
|u| (1 + s)
ds
ds
p−1
p−1
.
EIGENVALUE ESTIMATES — October 9, 2007
19
This proves the inequality with the claimed constant.
Lemma 6.5. If 1 ≤ d < 2 and β = 0, then (6.1) holds for q = ∞ with the constant
K(∞, 0, d) = (2d)d (2(d − 1))−2(d−1) (2 − d)−1 .
Proof. If d = 1 one has
2
|u(t)| ≤ 2
Z
∞
′
|u||u | ds ≤ 2
t
Z
∞
2
|u| ds
0
1/2 Z
∞
|u | ds
0
as claimed. If 1 < d < 2 then we estimate for any R > 0
Z R
Z ∞
′
′
2
|u||u | ds +
|u||u | ds
|u(t)| ≤ 2
Z
∞
=2
∞
0
′ 2 d−1
|u | s
0
+
Z
1/2
,
(6.3)
R
0
≤2
′ 2
Z
∞
0
ds
1/2
′ 2 d−1
|u | s
Z
R
1/2 Z
∞
kuk∞
ds
s
−d+1
ds
0
0
2 d−1
|u| s
1/2
ds
1/2
R
−d+1
!
1/2 "
kuk∞ (2 − d)−1/2 R(2−d)/2
|u′ |2 sd−1 ds
+
Z
0
∞
|u|2 sd−1 ds
1/2
#
R−d+1 .v
Choosing t such that u(t) = kuk∞ and optimizing with respect to R we find that
Z
d/2 Z
(2−d)/2
2
′ 2 d−1
2 d−1
kuk∞ ≤ K
|u | s
ds
|u| s
ds
with the constant as claimed. This implies (and, by a scaling argument, is actually equivalent
to) the assertion.
Lemma 6.6. If d = 1 and 0 < β ≤ 21 , then (6.1) holds for q = 2 with the constant
K(2, β, 1) = 2−2β (1 − 2β)2β−1 β −1 .
Proof. It suffices to prove the inequality
(1+2β)/2
(1−2β)/2 Z
Z
Z
2 d−1
′ 2
2 −1+2β
.
|v| s
ds
|v | ds
|v| s
ds ≤ K
(Actually, a scaling argument as in the proof of Theorem 6.1 below shows that this inequality
is equivalent – with the same constant – to the inequality (6.1).) Using (6.3) we estimate
for any R > 0
Z
Z R
2 −1+2β
2
|v| s
ds ≤ kvk∞
s−1+2β ds + kvk22 R−1+2β
0
≤β
−1
kvkkv ′ kR2β + kvk22 R−1+2β ,
and the claim follows by optimizing with respect to R.
Proof of Theorem 6.1. First assume that 1 < d ≤ 2 and 0 < β ≤ d−1
2 , or d > 2 and
d−1
d−2
≤
β
≤
.
The
assertion
(1)
has
been
proved
in
the
endpoint
cases
q = 2 and q = ∞
2
2
in Lemmas 6.3 and 6.4. Estimating
Z
Z
|u|q (1 + t)βq−1 dt ≤ sup |u|q−2 (1 + t)β(q−2)
|u|2 (1 + t)β2−1 dt
20
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
we obtain the assertion (1) also in the case 2 < q < ∞.
d
Next we prove the assertion (2). Let d ≥ 1, d−1
2 < β ≤ 2 . First assume that q = 2. If
d = 1, the inequality holds by Lemma 6.6. If d > 1 we put p : (2β − d + 1)−1 and apply
Hölder’s inequality to find
1
p−1 Z
Z
Z
p
p
2
d−1
2
d−2
2
2β−1
|u| (1 + t)
dt
≤
.
|u| (1 + t)
dt
|u| (1 + t)
dt ≤
Estimating the first factor on the right side using Lemma 6.3 with β ≤ d−1
2 we obtain the
d−1 −1
. We estimate
assertion in the case q = 2. Now let q = β − 2
Z
d−2β Z
q
2β−1
2
d−1 2β−d+1
2
d−1
|u| (1 + t)
dt ≤ sup |u| (1 + t)
|u| (1 + t)
dt .
The first factor on the right side is estimated using (6.3) if d = 1 and using Lemma 6.4
d−1 −1
. By Hölder’s
with β ≤ d−1
2 if d > 1. This proves the assertion in the case q β − 2
d−1 −1
.
inequality we obtain (2) for arbitrary 2 < q < β − 2
The assertion (3) was proved in Lemma 6.5.
To prove the negative results let 1 ≤ d ≤ 2 and assume that (6.1) holds for some β and
some 2 ≤ q ≤ ∞. We apply the inequality to the function u(t) = v(t/l), where v is a smooth
function with bounded support. Letting l → ∞ we obtain
1−θ
θ Z
Z
2/q
Z
2 d−1
′ 2 d−1
q βq−1
.
(6.4)
|v| s
ds
|v | s
ds
≤ K(q, β, d)
|v| s
ds
Note that v can be chosen non-zero in a neighborhood of the origin. We deduce that the
inequality can not hold for β < 0, and if q < ∞ then it can not hold for β = 0 either. This
proves assertion (4) and the first part of (5). It remains to prove that (6.1) or equivalently
(6.4) does not hold if d = 2, β = 0 and q = ∞. This follows by considering the sequence of
trial functions vn (s) := min{1, (log n − log s)/ log n} if s ≤ n and vn (s) = 0 for s > n.
d
Finally, to prove (6) let d ≥ 1 and d−1
2 < β ≤ 2 . Again we apply the inequality to the
function u(t) = v(t/l), where v is a smooth function with bounded support. As l → 0, the
left hand side decays like l2/q (resp. becomes constant when q = ∞) whereas the right hand
−1
side decays like l2β−d+1 . We conclude that the condition q ≤ β − d−1
is necessary for
2
(6.1) to hold.
7. Estimates for moments of eigenvalues
Our goal in this section will be to prove the Lieb-Thirring bounds in Theorem 2.18.
Throughout we will assume that g has power-like growth in the sense of (2.23) for some
d ≥ 1.
7.1. One-bound-state inequalities and duality. A first step towards Theorem 2.18 is
to prove that the lowest eigenvalue of the operator Ag − V can be estimated from below by
a weighted Lp -norm of the potential.
Theorem 7.1. Assume (2.23) for some d ≥ 1 and let a, γ ≥ 0. Then the inequality
Z
a
γ+ a+1
C = C(γ, a, d, c1 , c2 ),
sup spec (Ag − V )γ− ≤ C
V (t)+ 2 g(t) d−1 dt,
(7.1)
R+
holds for all V if and only if a and γ satisfy the assumptions of Theorem 2.18.
a+1
R
In the case γ = 0, inequality (7.1) means that if R+ V (t)+2 (1 + t)a dt < C −1 then
inf spec(Ag − V ) ≥ 0.
EIGENVALUE ESTIMATES — October 9, 2007
1
q
p
6
21
6
F (γ1 )
F (γ3 )
1
2
γ1
n
max 0,
γ3
γ2
o
d−2
2
1
F (γ2 )
d−1
2
d
2
min {1, d − 1}
β
a
Figure 1. Parameter range of the Sobolev interpolation inequalities. Here
F (1/q, β) = (q, (d − 1 − 2β)q + 2)/(q − 2) and F (γ1 ) = {(p, a) : p = (a +
1)/ min{2, d}}.
The proof of Theorem 7.1 is based on the following abstract duality result, which does
not use the explicit form of g.
Proposition 7.2. Assume that the parameters a > −1, γ ≥ 0 and p := γ +
d−2β
d
to the parameters 2 < q ≤ ∞, d−2
by
2 ≤ β < 2 and θ :=
2
p=
q
,
q−2
q=
2p
,
p−1
a=
(d − 1 − 2β)q + 2
,
q−2
β=
1+a
2
dp − 1 − a
,
2p
see Figure 1. Then the inequality (7.1) holds if and only if
1−θ
θ Z
Z
2/q
Z
βq−1
2
′ 2
q d−1
.
|u| g dt
|u | g dt
≤ K(q, β, g)
dt
|u| g
are related
(7.2)
(7.3)
for all u ∈ H 1 (R+ , g). In this case, the constants are related by
K(q, β, g) = L
q−2
q
θ −θ (1 − θ)θ−1
(7.4)
In the case q = ∞, (7.3) means
2β
sup |u|2 g d−1 ≤ Lθ −θ (1 − θ)θ−1
Z
|u′ |2 g dt
θ Z
|u|2 g dt
1−θ
.
for all u ∈ H 1 (R+ , g).
Proof of Proposition 7.2. Below we will only consider u ∈ H 1 (R+ , g) and V ≥ 0 such that
the right hand side of (7.1) is finite.
Equation (7.1) holds for all V if and only if
R ′2
R
Z
2/(2p−1−a)
a
|u | g dt − V |u|2 g dt
p d−1
R
≥
−
L
V
g
(7.5)
dt
|u|2 g dt
holds for all u and V . Write V = αW with α such that
Z
a
W p g d−1 dt = 1.
Thus (7.5) holds for all u and V if and only if
Z
Z
Z
q−2
1
2
2
sup α W |u| g dt − α 1−θ L q(1−θ) |u| g dt ≤ |u′ |2 g dt
α>0
(7.6)
(7.7)
22
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
holds for all u and all W obeying (7.6). By calculating the supremum we find that (7.8)
holds for all u and all W obeying (7.6) if and only if
Z
1−θ
θ Z
Z
Z
a
2
p d−1
2
′ 2
W |u| g dt : W g
sup
(7.8)
|u| g dt
|u | g dt
dt = 1 ≤ K
for all u. By duality
Z
Z
2/q
Z
βq−1
a
2
p d−1
q d−1
W |u| g dt :
W g
sup
.
|u| g
dt = 1 =
dt
Hence (7.8) holds for all u if and only if (7.3) holds for all u.
Proof of Theorem 7.1. Assumption (2.23) implies that Theorem 6.1 holds (with another constant) if (1 + t)d−1 is replaced by g. Simple arithmetic shows that if (q, β) and (p, a) are
related as in (7.2), then the allowed values (q, β) in Theorem 6.1 correspond to the allowed
values (p, a) in Theorem 2.18. In view of Proposition 7.2 we obtain the assertion of Theorem 7.1.
Remark 7.3. We claim that if the inequality
sup spec (Ag − V
)γ−
≤C
Z
γ+ 1+a
2
R+
V (t)+
g(t)b dt
(7.9)
holds for some γ ≥ 0, a ≥ 0, b ≥ 0 and all V , then one has necessarily b ≥ a/(d − 1).
Obviously, the inequality becomes weaker as b increases. This motivates why we restrict
ourselves to the case b = a/(d − 1) when considering the inequalities (2.24).
To prove the claim we apply a similar duality argument as in the proof of Proposition 7.2
and find that (7.9) is equivalent to
Z
θ Z
1−θ
2/q
Z
p−b
′ 2
2
q p−1
, u ∈ H 1 (R+ , g),
≤K
|u | g dt
|u| g dt
dt
|u| g
where p and q are as in that proposition and θ = (p − γ)/p. It follows from Remark 6.2 that
(d − 1)(p − b)/(p − 1) + 1 ≤ q(d − 2θ)/2. This means b ≥ a/(d − 1), as claimed.
7.2. Estimates in the case of a Dirichlet boundary condition. Here we will establish
the analog of Theorem 2.18 when a Dirichlet instead of a Neumann boundary condition is
imposed at the origin. More precisely we denote by AD the self-adjoint operator in L2 (R+ , g)
corresponding to the quadratic form (2.13) with form domain H01 (R+ , g) := {f ∈ H 1 (R+ , g) :
f (0) = 0}. In this case the conditions for the validity of a Lieb-Thirring inequality become
much simpler than in Theorem 2.18.
Theorem 7.4. Assume (2.23) for some d ≥ 1 and let a ≥ 0, γ > 0. Then the inequality
Z
γ+ a+1
γ
V (t)+ 2 (1 + t)a dt,
tr(AD − V )− ≤ L
L = L(γ, a, d, c1 , c2 ),
(7.10)
R+
holds for all V if and only if a, γ satisfy
γ≥
1−a
2
γ>0
if 0 ≤ a < 1,
if a ≥ 1.
We emphasize that we did not discuss the case γ = 0 in Theorem 7.4.
When proving Theorem 7.4 we will use a result from [8] and [9] concerning the operator
1
d2
− dr
2 − 4r 2 − W in L2 (R+ ) with a Dirichlet boundary condition at the origin.
EIGENVALUE ESTIMATES — October 9, 2007
Proposition 7.5. Let 0 ≤ a < 1 and γ ≥ 1−a
2 or a ≥ 1 and γ > 0, then
γ
Z
d2
1
γ+ 1+a
tr − 2 − 2 − W
W (r)+ 2 r a dr
≤ Cγ,a
dr
4r
R+
−
23
(7.11)
with a constant Cγ,a independent of W .
Before we can apply this estimate we have to replace the (possibly non-smooth) function
g by a smooth function with the same behavior at infinity. To this end we consider the
self-adjoint operator BD in L2 (R+ ) corresponding to the quadratic form
′ 2
Z u(t)
d−1
dt
bD [u] =
(1 + t)(d−1)/2 (1 + t)
R+
(7.12)
Z (d − 1)(d − 3)|u|2
′ 2
|u | +
=
dt
4(1 + t)2
R+
defined on H01 (R+ ). We prove now that the eigenvalues of AD − V can be estimated –
modulo a change in the coupling constant – from above and below by those of BD − V . A
similar idea was used in [13] to obtain Lieb-Thirring inequalities for Schrödinger operators
with background potentials.
Lemma 7.6. Assume (2.23) for some d ≥ 1 and put β := c2 /c1 . Then for any V ≥ 0 and
γ ≥ 0 we have
tr(BD − β −1 V )γ− ≤ tr(AD − V )γ− ≤ tr(BD − βV )γ− .
(7.13)
Proof. We shall prove that for any τ > 0
N (BD − β −1 V + τ ) ≤ N (AD − V + τ ) ≤ N (BD − βV + τ ).
(7.14)
This will imply the statement since
tr T−γ
=γ
Z
∞
τ γ−1 N (T + τ ) dτ.
0
To prove the second inequality in (7.14) suppose that
Z
Z
′ 2
2
|f | − V |f | g dt < −τ
R+
R+
|f |2 g dt
for some f ∈ H01 (R+ , g). Using (2.23) we conclude that
Z
Z
d−1
′ 2
2
|f ′ |2 − V |f |2 g dt
|f | − βV |f | (1 + t)
dt ≤
c1
R+
R+
Z
|f |2 g dt
≤ −τ
R+
Z
|f |2 (1 + t)d−1 dt .
≤ −τ c1
R+
It follows from Glazman’s lemma (see, e.g., [4, Thm. 10.2.3]) that
N (AD − V + τ ) ≤ N (ÃD − βV + τ ),
ÃD denotes the operator L2 (R+ , (1 + t)d−1 ) corresponding to the quadratic form
Rwhere
|f ′ |2 (1 + t)d−1 dt with a Dirichlet boundary condition. Since ÃD − βV in L2 (R+ , (1 + t)d−1 )
is unitarily equivalent to BD − βV in L2 (R+ ), we obtain the second inequality in (7.14).
The first one is proved similarly.
24
TOMAS EKHOLM, RUPERT L. FRANK, AND HYNEK KOVAŘÍK
Proof of Theorem 7.4. We may assume that V ≥ 0. We use the operator inequality
d2
1
d2
(d − 1)(d − 3)
−
≤
−
+
.
dr 2 4r 2
dr 2
4r 2
(Note also that the form domain of the operator on the LHS is strictly larger than H01 (R+ ).)
It follows that
γ
d2
1
γ
tr(BD − βV )− ≤ tr − 2 − 2 − βV
.
dr
4r
−
The result now follows from Proposition 7.5 and Lemma 7.6.
−
7.3. Putting it all together. Finally we give the
Proof of Theorem 2.18. The variational principle implies that the eigenvalues of the Dirichlet
and the Neumann problems interlace (see, e.g., [4, Thm. 10.2.5]). Hence
tr(A − V )γ− ≤ sup spec (A − V )γ− + tr(AD − V )γ− .
We estimate the first term on the right hand side via Theorem 7.1 (recall (2.23)) and the
second one via Theorem 7.4. This completes the proof of the ‘if’ part of the statement. The
‘only if’ statement follows from the ‘only if’ part of Theorem 7.1.
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J. Eur. Math. Soc., to appear. Preprint arXiv: math.SP/0611247.
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[13] R. L. Frank, B. Simon, T. Weidl, Eigenvalue bounds for perturbations of Schrödinger operators
and Jacobi matrices with regular ground states, Comm. Math. Phys., to appear. Preprint: arXiv:
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[15] D. Hundertmark, Some bound state problems in quantum mechanics. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, 463–496, Proc. Sympos.
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[16] D. Hundertmark, E. Lieb, L. Thomas, A sharp bound for an eigenvalue moment of the onedimensional Schrödinger operator, Adv. Theor. Math. Phys. 2 (1998), no. 4, 719–731.
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[17] D. Hundertmark, B. Simon, Lieb-Thirring inequalities for Jacobi matrices. J. Approx. Theory 118
(2002), no. 1, 106–130.
[18] V. Kostrykin and R. Schrader, Kirchhoff’s rules for quantum wires, J. Phys. A 32 (1999) 595–630.
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to appear. Preprint: arXiv: math-ph/0608013.
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S107–S128.
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35 (2002), no. 1, 101–121.
[23] A. Laptev, T. Weidl, Recent results on Lieb-Thirring inequalities. Journées “Équations aux Drives
Partielles” (La Chapelle sur Erdre, 2000), Exp. No. XX, 14 pp., Univ. Nantes, Nantes, 2000.
[24] E. H. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem. Geometry of the Laplace operator, Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc.,
Providence, R.I., 1980.
[25] E. H. Lieb, J. P. Solovej, J. Yngvason, Asymptotics of heavy atoms in high magnetic fields. II.
Semiclassical regions, Comm. Math. Phys. 161 (1994), no. 1, 77–124.
[26] E. H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics, 269–303. Princeton University Press, Princeton, NJ, 1976.
[27] K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees,
Proc. London Math. Soc. 80 (2000), no. 3, 690–724.
[28] K. Naimark and M. Solomyak, Geometry of the Sobolev spaces on the regular trees and Hardy’s
inequalities, Russ. J. Math. Phys. 8 (2001), no. 3, 322–335.
[29] B. Opic and A. Kufner, Hardy inequalities, Pitman Research Notes in Mathematics 219. Longman
Scientific & Technical, Harlow, 1990.
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Math. (Iz. VUZ) 20 (1976), 63–71.
[31] M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV, Academic press, New York
(1978).
[32] A. Sobolev and M. Solomyak, Schrödinger operators on homogeneous metric trees: spectrum in
gaps, Rev. Math. Phys. 14 (2002) 421–467.
[33] M. Solomyak, On the spectrum of the Laplacian on metric trees. Special section on quantum graphs,
Waves Random Media 14 (2004), no. 1, S155–S171.
[34] T. Weidl, On the Lieb-Thirring constants Lγ,1 for γ ≥ 1/2, Comm. Math. Phys. 178 (1996), no. 1,
135–146.
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Springer-Verlag, Berlin, 1987.
Tomas Ekholm, Centre for Mathematical Sciences, Lund University, Box 118, 22100 Lund,
Sweden
E-mail address: [email protected]
Rupert L. Frank, Department of Mathematics, Fine Hall, Princeton University, Princeton,
NJ 08544, USA
E-mail address: [email protected]
Hynek Kovařı́k, Department of Mathematics, Stuttgart University, Pfaffenwaldring 57,
70569 Stuttgart, Germany
E-mail address: [email protected]
Chapter 4
Spectral estimates in two
dimensions
4.1
Logarithmic Lieb-Thirring inequality
Published in Comm. Math. Phys. 275 (2007) 827–838.
197
Commun. Math. Phys. 275, 827–838 (2007)
Digital Object Identifier (DOI) 10.1007/s00220-007-0318-z
Communications in
Mathematical
Physics
Spectral Estimates for Two-Dimensional Schrödinger
Operators with Application to Quantum Layers
Hynek Kovařík, Semjon Vugalter, Timo Weidl
Institute of Analysis, Dynamics and Modeling, Universität Stuttgart, PF-80 11 40, D-70569 Stuttgart, Germany.
E-mail: [email protected]
Received: 21 December 2006 / Accepted: 11 January 2007
Published online: 16 August 2007 – © Springer-Verlag 2007
Abstract: A logarithmic type Lieb-Thirring inequality for two-dimensional Schrödinger
operators is established. The result is applied to prove spectral estimates on trapped
modes in quantum layers.
1. Introduction
It is well known that the sum of the moments of negative eigenvalues −λ j of a
2
one-dimensional Schrödinger operator − ddx 2 − V can be estimated by
j
γ
λ j ≤ L γ ,1
1
R
V+ (x)γ + 2 d x, γ ≥
1
,
2
(1)
where L γ ,1 is a constant independent of V , see [10, 16]. For γ = 21 this bound has
the correct weak coupling behavior, see [13], and it also shows the correct Weyl-type
asymptotics in the semi-classical limit. Moreover, (1) fails to hold whenever γ < 21 . The
case γ = 21 therefore represents a certain borderline inequality in dimension one.
The situation is much less satisfactory in dimension two. The corresponding
two-dimensional Lieb-Thirring bound
j
γ
λj
= tr (− −
γ
V )−
≤ L γ ,2
R2
V+ (x)γ +1 d x
(2)
holds for all γ > 0, [10]. Dimensional analysis shows that here the borderline should
be γ = 0. However, (2)
fails for γ = 0, because − − V has at least one negative
eigenvalue whenever V ≥ 0, see [13]. In addition, it was shown in [13] that if V
828
H. Kovařík, S. Vugalter, T. Weidl
decays fast enough, the operator − − αV has for small α only one eigenvalue which
goes to zero exponentially fast:
λ1 ∼ e−4π(α
V )−1
, α → 0.
(3)
It follows from (3) that the optimal behavior for α → 0 cannot be reached in the
power-like scale (2), no matter how small γ is, since the l.h.s. decays faster than any
power of α. This means that in order to obtain a Lieb-Thirring type inequality with the
optimal behavior in the weak coupling limit, one should introduce a different scale on
the l.h.s. of (2).
In the present paper we want to find a two-dimensional analog of the one-dimensional
borderline inequality, which corresponds to γ = 21 in (1). In other words, we want to
establish an inequality with the r.h.s. proportional to V and with the correct order of
asymptotics in the weak and strong coupling regime. Obviously, we have to replace the
power function on the l.h.s. of (2) by a new function F(λ), which will approximate
identity as close as possible. On the other hand, since − − V has always at least one
eigenvalue, it is necessary that F(0) = 0. Moreover, Eq. (3) shows that F should grow
from zero faster than any power of λ, namely as | ln λ|−1 . This leads us to define the
family of functions Fs : (0, ∞) → (0, 1] by

0 < t ≤ e−1 s −2 ,
 | ln ts 2 |−1
∀s > 0
Fs (t) :=
(4)

1
t > e−1 s −2 .
Notice that each Fs is non-decreasing and continuous and that Fs (t) → 1 point-wise
as s → ∞. Hence our
goal is to establish an appropriate estimate on the regularized
counting functionn
j Fs (λ j ) for large values of the parameter s.
Our main results are formulated in the next section. It turns out that
j Fs (λ j )
can be estimated by a sum of two integrals, one of which includes a local logarithmic
weight, see Theorem 1. The inequality (8) established in Theorem 1 has the correct
behavior for weak as well as for strong potentials, see Remark 1. We also show that
the logarithmic weight in (8) cannot be removed, see Remark 2. Moreover, in Corollary
1 we obtain individual estimates on eigenvalues of Schrödinger operators with slowly
decaying potentials. The proof of the main result, including two auxiliary lemmata, is
then given in Sect. 3. In Remark 4 we give some numerical estimates on the constants in
the inequality (9). In the closing Sect. 4 we apply Theorem 1 to analyze discrete spectrum
of a Schrödinger operator corresponding to quantum layers. The result established in
Sect. 4 may be regarded as a two-dimensional analog of Lieb-Thirring inequalities on
trapped modes in quantum waveguides obtained in [6].
2. Main Results
For a given V we define the Schrödinger operator
− − V in L 2 (R2 )
as the Friedrich extension of the operator associated with the quadratic form
|∇u|2 − V |u|2 d x on C0∞ (R2 ),
Q V [u] =
(5)
(6)
R2
provided Q V is bounded from below. Throughout the paper we will suppose that V
satisfies
Spectral Estimates for 2-D Schrödinger Operators
829
Assumption A. The function V (x) is such that σess (− − V ) = [0, ∞).
The following notation will be used in the text. Given a self-adjoint operator T , the
number of negative eigenvalues, counting their multiplicity, of T to the left of a point
−ν is denoted by N (ν, T ). The symbol R+ stands for the set (0, ∞). Moreover, as in
[7] we define the space L 1 (R+ , L p (S1 )) in polar coordinates (r, θ ) in R2 , as the space
of functions f such that
1/ p
∞ 2π
p
f L 1 (R+ ,L p (S1 )) :=
| f (r, θ )| dθ
r dr < ∞.
(7)
0
0
Finally, given s > 0 we denote B(s) := {x ∈ R2 : |x| < s}. We then have
1 (R2 , | ln |x|| d x). Assume that V ∈ L 1 (R , L p (S1 ))
Theorem 1. Let V ≥ 0 and V ∈ L loc
+
for some p > 1. Then the quadratic form (6) is bounded from below and closable.
The negative eigenvalues −λ j of the operator associated with its closure satisfy the
inequality
Fs (λ j ) ≤ c1 V ln(|x|/s) L 1 (B(s)) + c p V L 1 (R+ ,L p (S1 ))
(8)
j
for all s ∈ R+ . The constants c1 and c p are independent of s and V . In particular, if
V (x) = V (|x|), then there exists a constant c4 , such that
Fs (λ j ) ≤ c1 V ln(|x|/s) L 1 (B(s)) + c4 V L 1 (R2 )
(9)
j
holds true for all s ∈ R+ .
Remark 1. Notice that the r.h.s. of (8) has the right order of asymptotics in both weak
and strong coupling limits. Indeed, replacing V by αV and assuming that V ∈ L 1 (R2 ,
(| ln |x|| + 1) d x) it can be seen from the definition of Fs that
Fs (λ j ) ∼ α, α → 0 ∨ α → ∞.
j
For α → 0 this follows from (3). For α → ∞ the behavior of j Fs is governed by
the Weyl asymptotics for the counting function:
N (e−1 s −2 , − − αV ) ≤
Fs (λ j ) ≤ N (0, − − αV ).
(10)
j
The latter is linear in α when α → ∞ provided V ∈ L 1 (R2 , (| ln |x|| + 1) d x), see also
Remark 5.
Remark 2. We would like to emphasize that
j Fs (λ j ) cannot be estimated only in
terms of V L 1 (R2 ) . In particular, the logarithmic term in (8) and (9) cannot be removed. This is due to the fact that there exist potentials V ∈ L 1 (R2 ) with a strong local
singularity, such that the semi-classical asymptotics of N (ν, − − V ) is non-Weyl for
any ν > 0, [2]. Namely if we define
Vσ (x) = r −2 | ln r |−2 | ln | ln r ||−1/σ , r < e−2 , σ > 1,
Vσ (x) = 0,
r ≥ e−2 ,
(11)
830
H. Kovařík, S. Vugalter, T. Weidl
where r = |x|, then Vσ ∈ L 1 (R2 ) for all σ > 1, but
N (ν, − − αVσ ) ∼ α σ
α → ∞, ∀ ν > 0,
(12)
see [2, Sect. 6.5]. If (9) were true with the logarithmic factor removed, it would be in
obvious contradiction with (10) and (12). Moreover, the asymptotics (12) remain valid
also if the singularity of V is not placed at zero, but at some other point. This shows that
the condition p > 1 in Theorem 1 is necessary.
Remark 3. The non-Weyl asymptotics of N (0, − − αV ) can also occur for potentials
which have no singularities, but which decay at infinity too slowly, so that the associated
eigenvalues accumulate at zero. For example, if
Vσ (x) = (θ ) r −2 (ln r )−2 (ln ln r )−1/σ , r > e2 , σ > 1,
Vσ (x) = 0,
r ≤ e2 ,
(13)
then
N (0, − − αVσ ) ∼ α σ ,
see [2]. In this case, however, Theorem 1 says that the eigenvalues accumulating at zero
are small enough so that their total contribution to j Fs (λ j ) grows at most linearly in
α. More exactly, inequality (8) gives the following estimate:
Corollary 1. Let ∈ L p (0, 2π ) for some p > 1. Let V satisfy the assumptions of
Theorem 1 and suppose that
V (x) − Vσ (x) = o Vσ|| (x) , |x| → ∞,
where Vσ (x) is defined by (13). Denote n(α) = N (0, − − αV ) and let −λn(α) be the
largest eigenvalue of − − αV . Then, for any fixed s > 0 there exists a constant cs > 0
such that for α large enough we have
λn(α) ≤ s −2 exp(−cs α σ −1 ).
Proof. Inequality (8) shows that
implies
j
(14)
Fs (λ j ) ≤ cs′ α for some cs′ . In particular, this
j Fs (λ j ) ≤ cs′ α, ∀ j.
(15)
On the other hand, from [2, Prop. 6.1] it follows that n(α) ≥ c̃ α σ for some c̃ and α large
enough. An application of the inequality (15) with j = n(α) then yields (14). Analogous
estimates for λn(α)−k , k ∈ N can be obtained by an obvious modification. ⊓
⊔
Spectral Estimates for 2-D Schrödinger Operators
831
3. Proof of Theorem 1
We prove the inequality (8) for continuous potentials with compact support. The general
case then follows by approximating V by a sequence of continuous compactly supported
functions and using a standard limiting argument in (8).
As usual in the borderline situations, the method of [10] cannot be directly applied
and a different strategy is needed. We shall treat the operator − − V separately on the
space of spherically symmetric functions in L 2 (R2 ) and on its orthogonal complement.
To this end we define the corresponding projection operators:
2π
1
(Pu)(r ) =
u(r, θ ) dθ, Qu = u − Pu, u ∈ L 2 (R2 ).
2π 0
Since P and Q commute with −, the variational principle says that for each a > 1 the
operator inequality
− − V ≥ P (− − (1 + a −1 ) V ) P + Q (− − (1 + a) V ) Q
(16)
holds. Let us denote by −λ Pj and −λ Qj the non decreasing sequences of negative eigenvalues of the operators P (− − (1 + a −1 ) V ) P and Q (− − (1 + a V ) Q respectively.
Clearly we have
Q
Fs (λ j ) ≤
Fs (λ Pj ) +
Fs (λ j ).
(17)
j
j
j
We are going to find appropriate bounds on the two terms on the r.h.s. of (17) separately.
First we note that P (− − (1 + a −1 ) V ) P is unitarily equivalent to the operator
h=−
d2
1
− 2 − W (r ) = h 0 − W (r ) in L 2 (R+ )
dr 2
4r
with the Dirichlet boundary condition at zero and with the potential
1 + a 2π
W (r ) =
V (r, θ ) dθ.
2πa 0
More precisely, h is associated with the closure of the quadratic form
q[ϕ] =
|ϕ ′ |2 − W |ϕ|2 r dr on C0∞ (R+ ).
(18)
(19)
(20)
R+
We start with the estimate on the lowest eigenvalue of h.
Lemma 1. Let V be continuous and compactly supported and let W be given by (19).
Denote by −λ1P the lowest eigenvalue of the operator h. Then there exists a constant c2 ,
independent of s, such that
∞
Fs (λ1P ) ≤ c2
W (r ) r 1 + χ(0,s) (r ) | ln r/s| dr
(21)
0
holds true for all s ∈ R+ .
832
H. Kovařík, S. Vugalter, T. Weidl
Proof. From the Sturm-Liouville theory we find the Green function of the operator h 0
at the point −κ 2 :
√
0 ≤ r ≤ r ′ < ∞,
 rr ′ I0 (κr ) K 0 (κr ′ )
′
G 0 (r, r , κ) := √

rr ′ I0 (κr ′ ) K 0 (κr )
0 ≤ r ′ < r < ∞,
where I0 , K 0 are the modified Bessel functions, see [1]. The Birman-Schwinger principle
tells us that if for a certain value of κ the trace of the operator
√
√
K (κ) := W (h 0 + κ 2 )−1 W
is less than or equal to 1, then the inequality λ1P ≤ κ 2 holds. Taking into account the
continuity of W , this implies
∞
P
P
λ1 r K 0
λ1 r W (r ) dr ≥ 1.
(22)
r I0
0
Now we introduce the substitutions τ = s λ1P , t = s −1r and recall that I0 (0) = 1
while K 0 has a logarithmic singularity at zero, see [1, Chap. 9]. We thus find out that
F1 τ 2 I0 (τ t) K 0 (τ t) ≤ c2 1 + χ(0,1) (t) | ln t| , ∀τ ≥ 0,
where c2 is a suitable constant independent of τ . Here we have used the fact that
|I0 (z) K 0 (z)| ≤ const ∀ z ≥ 1,
(23)
see [1]. Numerical analysis gives c2 ∼
= 0.844. Finally, we multiply both sides of inequality (22) by Fs (λ1P ) and note that
Fs (λ1P ) = Fs τ 2 /s 2 = F1 τ 2 .
⊓
⊔
The proof is complete.
Next we estimate the higher eigenvalues of h.
Lemma 2. Under the assumptions of Lemma 1 there exists a constant c3 such that
s
∞
P
Fs (λ j ) ≤
W (r ) r |ln r/s| dr + c3
W (r ) r dr, ∀ s ∈ R+ .
j≥2
0
s
Proof. Let us introduce the auxiliary operator
hd = −
d2
1
− 2 − W (r ) in L 2 (R+ )
2
dr
4r
(24)
subject to the Dirichlet boundary conditions at zero and at the point s. Let −µ j be the
non-decreasing sequence of negative eigenvalues of h d . Since imposing the Dirichlet
boundary condition at s is a rank one perturbation, it follows from the variational principle
that
Fs (λ Pj ) ≤
Fs (µ j ).
(25)
j≥2
j≥1
Spectral Estimates for 2-D Schrödinger Operators
833
Moreover, h d is unitarily equivalent to the orthogonal sum h 1 ⊕ h 2 , where
1
d2
− 2 − W (r ) in L 2 (0, s),
dr 2
4r
d2
1
h 2 = h 2,0 − W (r ) = − 2 − 2 − W (r ) in L 2 (s, ∞)
dr
4r
h 1 = h 1,0 − W (r ) = −
with Dirichlet boundary conditions at 0 and s. Keeping in mind that Fs ≤ 1 we will
estimate (25) as follows:
Fs (µ j ) ≤ N (0, h 1 ) +
Fs (µ′j ),
(26)
j
j
where −µ′j are the negative eigenvalues of h 2 . To continue we calculate the diagonal
elements of the Green functions of the free operators h 1,0 and h 2,0 . Similarly as in the
proof of Lemma 1 we get
G 1 (r, r, κ) = r I0 (κr ) K 0 (κr ) + βs−1 (κ)I0 (κr )
0 ≤ r ≤ s,
(27)
s ≤ r < ∞,
G 2 (r, r, κ) = r K 0 (κr ) (I0 (κr ) + βs (κ)K 0 (κr ))
where
βs (κ) = −
I0 (κs)
.
K 0 (κs)
The Birman-Schwinger principle thus gives us the following estimates on the number
of eigenvalues of h 1 and h 2 to the left of the point −κ 2 :
s
∞
N (κ 2 , h 1 ) ≤
G 1 (r, r, κ) W (r ) dr, N (κ 2 , h 2 ) ≤
G 2 (r, r, κ) W (r ) dr. (28)
0
s
Passing to the limit κ → 0 and using the asymptotic behavior of the Bessel functions I0
and K 0 , [1], we find out that for any fixed r holds the identity
lim G 1 (r, r, κ) = lim G 2 (r, r, κ) = r |ln r/s| .
κ→0
κ→0
(29)
The assumption on W and the dominated convergence theorem then allow us to interchange the limit κ → 0 with the integration in (28) to obtain
s
N (0, h 1 ) ≤
r |ln r/s| W (r ) dr.
(30)
0
This estimates the first term in (26). In order to find an upper bound on the second term
in (26), we employ the formula
∞
′
Fs (µ j ) =
Fs′ (t) N (t, h 2 ) dt,
(31)
j
0
see [10]. Using (28), the substitution t → t 2 and the Fubini theorem we get
e−1/2 s −1
G 2 (r, r, t)
1 ∞
′
Fs (µ j ) ≤
dt dr.
W (r )
2 s
t (ln ts)2
0
j
834
H. Kovařík, S. Vugalter, T. Weidl
In view of (27) it suffices to show that the integral
e−1/2 s −1
0
K 0 (tr ) (I0 (tr ) + βs (t)K 0 (tr ))
dt
t (ln ts)2
(32)
is uniformly bounded for all s > 0 and r ≥ s. The substitutions r = sy, t = τ/s
transform (32) into
g(y) :=
e−1/2
0
K 0 (τ y) (I0 (τ y) + β1 (τ )K 0 (τ y))
dτ,
τ (ln τ )2
y ∈ [1, ∞).
(33)
Since g is continuous, due to the continuity of Bessel functions, and g(1) = 0, it is
enough to check that g(y) remains bounded as y → ∞. Moreover, the inequality
(u, (h 2,0 + t1 )−1 u) ≤ (u, (h 2,0 + t2 )−1 u) ∀ 0 ≤ t2 ≤ t1 , ∀ u ∈ L 2 (s, ∞)
shows that G 2 (r, r, t), the diagonal element of the integral kernel of (h 2,0 + t 2 )−1 , is non
increasing in t for each r ≥ s. Equations (27) and (29) then imply
0
y −1
K 0 (τ y) (I0 (τ y) + β1 (τ )K 0 (τ y))
dτ ≤ ln y
τ (ln τ )2
y −1
0
dτ
= 1.
τ (ln τ )2
On the other hand, when τ ∈ [y −1 , e−1/2 ], it can be seen from (23) and from the behavior
of I0 , K 0 in the vicinity of zero, see [1], that
|K 0 (τ y) (I0 (τ y) + β1 (τ )K 0 (τ y))| ≤ const
uniformly in y. Equation (31) thus yields
∞
′
Fs (µ j ) ≤ c3
W (r ) r dr ∀ s ∈ R+ ,
s
j
where c3 is independent of s. Numerical analysis shows that c3 ∼
= 0.7. Together with
(25), (26) and (30) this completes the proof. ⊓
⊔
From Eq. (19), Lemma 1 and Lemma 2 we conclude that
1+a
1+a
c2 + 1 V ln(|x|/s) L 1 (B(s)) +
c3 V L 1 (R2 ) .
Fs (λ Pj ) ≤
2πa
2πa
j
Let us now turn to the second term on the r.h.s. of (17). The key ingredient in estimating
this contribution will be the result of Laptev and Netrusov obtained in [7]. We make use
of the estimate
Q
Fs (λ j ) ≤ N (0, Q(− − (1 + a) V )Q)
j
and of the Hardy-type inequality
Q (−) Q ≥ Q
1
Q,
|x|2
(34)
Spectral Estimates for 2-D Schrödinger Operators
835
which holds in the sense of quadratic forms on C0∞ (R2 ), see [2]. For any ε ∈ (0, 1) we
thus get the lower bound
1
1+a
ε
V Q, (35)
Q (− − (1 + a) V ) Q ≥ (1 − ε) Q − +
−
1 − ε |x|2
1−ε
which implies
N (0, Q (− − (1 + a) V ) Q) ≤ N 0, − +
1
ε
1+a
V
.
−
1 − ε |x|2
1−ε
The last quantity can be estimated using [7, Thm.1.2], which says that
1
ε
1+a
N 0, − +
V
≤ c̃ p V L 1 (R+ ,L p (S1 ))
−
1 − ε |x|2
1−ε
(36)
(37)
for some constant c̃ p that also depends on ε and a. In order to conclude the proof of (8)
we note that by the Hölder inequality
V L 1 (R2 ) ≤ const V L 1 (R+ ,L p (S1 )) .
To show that the quadratic form (6) is semi-bounded from below we note that inequality (8) says that there are only finitely many eigenvalues of − − V below −e−1 s −2 .
Let −V be the minimum of those. Then
Q V [u] ≥ −V u L 2 (R2 ) ∀ u ∈ C0∞ (R2 ).
The proof of Theorem 1 is now complete.
Remark 4. The constant c p in Theorem 1 depends on p and generically goes to infinity as
p → 1, see Remark 2. However, for spherically symmetric potentials we have c p = c4 ,
which is independent of p, see (9). In this case we can use the result of [8], see also [3],
to get an upper bound on c4 . Taking into account the numerical values of c2 and c3 we
set a = 1 and optimize w.r.t. ε. This gives c1 ∼
= 1.08.
= 1.27 and c4 ∼
Remark 5. As a corollary of the proof of Theorem 1 we immediately obtain
N (0, − − V ) ≤ 1 + const V ln |x/s| L 1 (R2 ) + V L 1 (R+ ,L p (S1 )) ,
(38)
which agrees with [14, Thm.3]. Estimates on N (0, − − V ), different from (38), including logarithmic weights have been obtained earlier in [11, 15]. For spherically symmetric potentials (38) reduces to the inequality established, with explicit constants, already
in [3].
Remark 6. Lieb-Thirring inequalities for the operator h = h 0 − W in the form
γ + 1+a
γ
tr (h 0 − W )− ≤ Cγ ,a
W (r )+ 2 r a dr, γ > 0, a ≥ 1
R+
have been recently established in [5].
836
H. Kovařík, S. Vugalter, T. Weidl
4. Application
In this section we consider a model of quantum layers. It concerns a conducting plate
= R2 × (0, d) with an electric potential V . We will consider the shifted Hamiltonian
HV = − − V −
π2
d2
in L 2 (),
(39)
with Dirichlet boundary conditions at ∂, which is associated with the closed quadratic
form
π2 2
2
2
|∇u| − V |u| − 2 |u| d x on H01 ().
(40)
d
We assume that for each x3 ∈ (0, d) the function V (·, ·, x3 ) satisfies Assumption A.
Without loss of generality we assume that V ≥ 0, otherwise we replace V by its positive
part.
The essential spectrum of the Operator HV covers the half line [0, ∞). Let us denote
by −λ̃ j the non-decreasing sequences of negative eigenvalues of HV . For the sake of
brevity we choose s = 1 and prove
Theorem 2. Assume that V ∈ L 3/2 () and that
π x 2 d
3
Ṽ (x1 , x2 ) =
d x3
V (x1 , x2 , x3 ) sin2
d 0
d
satisfies the assumptions of Theorem 1 for some p > 1. Then there exist positive constants
C1 , C2 , C3 ( p) such that
F1 (λ̃ j ) ≤ C1 Ṽ ln(x12 + x22 ) L 1 (B(1)) + C3 ( p) Ṽ L 1 (R+ ,L p (S1 ))
j
+ C2 V 3/2 L 1 () .
(41)
Remark 7. Notice that (41) has the right asymptotic behavior in both weak and strong
coupling limits. Namely, in the weak coupling limit the r.h.s. is dominated by the term
linear in V , while in the strong coupling limit the term proportional to V 3/2 prevails.
In this sense our result is similar to the Lieb-Thirring inequalities on trapped modes in
quantum wires obtained in [6].
Proof of Theorem 2. Let νk = k 2 π 2 /d 2 , k ∈ N be the eigenvalues of the Dirichlet
Laplacian on (0, d) associated with the normalized eigenfunctions
k π x3
2
φk (x3 ) =
.
sin
d
d
Moreover, define
R = (φ1 , ·) φ1 , S = I − R.
By the same variational argument used in the previous section we obtain the inequality
HV ≥ R (− − ν1 − 2V ) R + S (− − ν1 − 2V ) S.
(42)
Spectral Estimates for 2-D Schrödinger Operators
837
The latter implies
F1 (λ̃ j ) ≤
F1 (µ̃ j ) + N (0, S (− − ν1 − 2V ) S),
j
(43)
j
where −µ̃ j are the negative eigenvalues of R (− − ν1 − 2V ) R. Since
R (− − ν1 − 2V ) R = (−∂x21 − ∂x22 − 2 Ṽ ) ⊗ R,
the first term on the r.h.s. of (43) can be estimated using (8) as follows:
F1 (µ̃ j ) ≤ C1 Ṽ1 ln(x12 + x22 ) L 1 (R2 ) + C3 ( p) Ṽ L 1 (R+ ,L p (S1 )) .
(44)
j
As for the second term, we note that
S (−∂x23
∞
∞
ν2 − ν1
νk (φk , ·) φk
− ν1 ) S =
(νk − ν1 ) (φk , ·) φk ≥
ν2
k=2
k=2
3
= S (−∂x23 ) S
4
holds true in the sense of quadratic forms on C0∞ (0, d), which implies the estimate
3
8
S (− − ν1 − 2V ) S ≥ S − − V S.
4
3
Using the variational principle and the Cwickel-Lieb-Rosenblum inequality, [4, 9, 12],
we thus arrive at
8
N (0, S (− − ν1 − 2V ) S) ≤ N 0, − − V ≤ C2
V 3/2 .
3
In view of (43) this concludes the proof.
⊓
⊔
Acknowledgement. We would like to thank Elliott Lieb for useful comments. The support from the DFG grant
WE 1964/2 is gratefully acknowledged.
References
1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Washington, DC: National Bureau
of Standards, 1964
2. Birman, M.S., Laptev, A.: The negative discrete spectrum of a two-dimensional Schrödinger operator.
Comm. Pure and Appl. Math. XLIX, 967–997 (1996)
3. Chadan, K., Khuri, N.N., Martin, A., Wu, T.T.: Bound states in one and two spatial dimensions. J. Math.
Phys. 44, 406–422 (2003)
4. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger
operators. Ann. of Math. 106, 93–100 (1977)
5. Ekholm, T., Frank, R.L.: Lieb-Thirring inequalities on the half-line with critical exponent.
http://arxiv.org/list/math.SP/0611247, 2006
6. Exner, P., Weidl, T.: Lieb-Thirring inequalities on trapped modes in quantum wires. XIIIth International
Congress on Mathematical Physics (London, 2000), Boston, MA: Int. Press, 2001, pp. 437–443
7. Laptev, A., Netrusov, Y.: On the negative eigenvalues of a class of Schrödinger operators, In: Diff. operators
and spectral theory. Am. Math. Soc. Transl. 2, 189, 173–186 (1999)
838
H. Kovařík, S. Vugalter, T. Weidl
8. Laptev, A.: The negative spectrum of a class of two-dimensional Schrödinger operators with spherically
symmetric potentials. Func. Anal. Appl. 34, 305–307 (2000)
9. Lieb, E.: Bound states of the Laplace and Schrödinger operators. Bull. Amer. Math. Soc. 82, 751–753
(1976)
10. Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian
and their relation to Sobolev inequalities. Studies in Mathematical Physics. Princeton, NJ: Princeton
University Press, 1976, pp. 269–303
11. Newton, R.G.: Bounds on the number of bound states for the Schrödinger equation in one and two
dimensions. J. Op. Theory 10, 119–125 (1983)
12. Rosenblum, G.V.: Distribution of the discrete spectrum of singular differential operators (in Russian),
Izv. Vassh. Ucheb. Zaved. Matematika 1, 75–86 (1976); English transl. Soviet Math. 20, 63–71 (1976)
13. Simon, B.: The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Ann.
of Phys. 97, 279–288 (1976)
14. Solomyak, M.: Piecewise-polynomial approximation of functions from H l ((0, 1)d ), 2l = d, and
applications to the spectral theory of the Schrödinger operator. Israel J. of Math. 86, 253–275 (1994)
15. Stoiciu, M.: An estimate for the number of bound states of the Schrödinger operator in two
dimensions. Proc. of AMS 132, 1143–1151 (2003)
16. Weidl, T.: On the Lieb-Thirring constants L γ ,1 for γ ≥ 1/2. Commun. Math. Phys. 178(1), 135–146
(1996)
Communicated by B. Simon
4.2
Applications to quantum layers
To appear in J. Math. Anal. Appl.
205
Estimates on trapped modes in deformed
quantum layers
Hynek Kovařı́k and Semjon Vugalter
Institute of Analysis, Dynamics and Modeling, Universität Stuttgart, PF 80 11
40, D-70569 Stuttgart, Germany.
Abstract
We use the logarithmic Lieb-Thirring inequality for two-dimensional
Schrödinger operators and establish estimates on trapped modes in
geometrically deformed quantum layers.
1
Introduction
Trapped modes in quantum layers and waveguides have been intensively
studied in the last decades, see [1, 2, 3, 4, 6, 9] and references therein. In
these papers it has been shown that a suitable geometrical perturbation of a
waveguide (or a layer) Ω, such as local enlargement or bending, induces the
existence of discrete eigenvalues Ej of the corresponding Laplace operator
−∆Ω
in L2 (Ω)
with Dirichlet boundary conditions. These eigenvalues represent the so
called trapped modes, which are the main objects of our interest. For mildly
deformed waveguides and layers the corresponding weak coupling behaviour
of such eigenvalues has been established in [1, 2, 3, 4].
The next step in the analysis of the above mentioned eigenvalues consist
of deriving suitable spectral estimates. In other words, one would like to
know not only that these eigenvalues exist, but also in which way they are
linked to the deformation of Ω, i.e. how the distance of Ej to the essential
spectrum of −∆Ω depends on the perturbation. Such a connection can
be formulated in terms of certain Lieb-Thirring type inequalities, which
estimate the sums
X
|E − Ej |γ ,
E := inf σess (−∆Ω ) , γ ≥ 0 .
(1)
j
1
In the case in which Ω is a quantum waveguide, these estimates were proved
in [8] for potential type perturbations and in [5] for geometrical perturbations and perturbations of the boundary conditions. In the case of a quantum layer with a potential perturbation, the corresponding inequality was
recently obtained in [10]. All these estimates have the right order of asymptotics for weak perturbations, i.e. the respective upper bounds on the sum
(1) reflect the correct weak coupling behaviour established in [1, 2, 3, 11].
The aim of the present paper is to extend these results also to the case
of a geometrical deformation of a quantum layer. We note that in the case
of quantum waveguides the key ingredient of the proof of an estimate, which
has the correct asymptotical behaviour, was the Lieb-Thirring inequality for
one-dimensional Schrödinger operators with the critical power γ = 12 proved
in [12]. Since a layer might be considered as a two-dimensional analog of
a waveguide, the key ingredient of our proof will be the corresponding logarithmic critical Lieb-Thirring inequality for two-dimensional Schrödinger
operators, which was recently established in [10]. Therefore we first briefly
recall the result of [10]; see Theorem 1. In section 3 we then show how the
problem can be reduced to the spectral analysis of certain two-dimensional
Schrödinger operator with the effective potential induced by the geometrical
deformation of the layer.
The following notation will be adopted in the text. Given a Hilbert space
H and a self-adjoint operator T in H we denote by NH (T ) the number of
negative eigenvalues of T , counting their geometrical multiplicities. When
necessary we will use the symbols ∆x,y , ∇x,y etc. in order to specify in which
variables the respective operators act.
2
2.1
Preliminaries
Quantum layers
A quantum layer may be represented by an open domain Ω = R2 × (0, d),
more precisely Ω := {x, y, z ∈ R3 : 0 < z < d}, where d is the width of Ω.
It will be convenient to work with the shifted Laplace operator
π2
in L2 (Ω)
(2)
d2
with the Dirichlet boundary conditions at ∂Ω. The operator A is associated
with closed quadratic form
Z π2 2
2
dxdydz
(3)
Q[u] =
|∇u| − 2 |u|
d
Ω
A = −∆Ω −
2
with the form domain H01 (Ω). It can be easily verified that
σess (A) = [0, ∞),
σd (A) = ∅ .
As noted in [1], a local enlargement of the width of the layer will not affect
the essential spectrum of A, but will lead to the existence of negative discrete
eigenvalues of A. To find a suitable spectral estimate on these eigenvalues
we need the two-dimensional logarithmic Lieb-Thirring inequality, which we
formulate in the next section.
2.2
Two-dimensional Lieb-Thirring inequality
Consider the Schrödinger operator
−∆ − V
in
L2 (R2 ) ,
(4)
where V is a potential function decaying at infinity such that σess (−∆−V ) =
[0, ∞). Denote by −λj the negative eigenvalues of −∆ − V and introduce
the family of functions Fs : (0, ∞) → (0, 1] defined by

0 < t ≤ e−1 s−2 ,
 | ln ts2 |−1
∀s > 0
Fs (t) :=
(5)

1
t > e−1 s−2 .
An upper bound on the sum
X
Fs (λj )
j
in terms of intergals of V has been recently found in [10]. Its formulation
requires some additional notation. The space L1 (R+ , Lp (S1 )) is defined as
the space of functions f such that
1/p
Z ∞ Z 2π
p
r dr < ∞ ,
(6)
|f (r, θ)| dθ
kf kL1 (R+ ,Lp (S1 )) :=
0
0
where (r, θ) are the polar coordinates in R2 . Moreover, given an s > 0 we
introduce B(s) := {x ∈ R2 : |x| < s}. The result of [10] then reads as
follows:
Theorem 1. Let V ≥ 0 and V ∈ L1loc (R2 , | ln |x|| dx). Assume that V ∈
L1 (R+ , Lp (S1 )) for some p > 1. Then the eigenvalues −λj satisfy the inequality
X
Fs (λj ) ≤ c1 kV ln(|x|/s)kL1 (B(s)) + cp kV kL1 (R+ ,Lp (S1 ))
(7)
j
3
for all s ∈ R+ . The constants c1 and cp are independent of s and V .
In particular, if V (x) = V (|x|), then there exists a constant c4 , such that
X
Fs (λj ) ≤ c1 kV ln(|x|/s)kL1 (B(s)) + c4 kV kL1 (R2 )
(8)
j
holds true for all s ∈ R+ .
Note that for weak potentials V the estimate (7) reflects the exponential
asymptotical behaviour of the lowest eigenvalue of −∆ − V established in
[11]. Since the behaviour of weakly coupled eigenvalues in a layer is essentially two-dimensional, the corresponding asymptotics for weakly deformed
layers is again of the exponential type, see [1]. Our goal thus is to find a
similar upper bound for geometrical induced eigenvalues in quantum layers.
3
A layer with a geometrical perturbation
Here we apply Theorem 1 to obtain the estimates on the discrete eigenvalues
of the Dirichlet Laplacian in a layer whose width is locally enlarged;
Ωf := {x, y, z ∈ R3 : 0 < z < d + f (x, y)},
where f : R2 → [0, ∞). We consider the shifted Laplace operator
Af = −∆Ωf −
π2
d2
in
L2 (Ωf )
(9)
with the Dirichlet boundary conditions at ∂Ωf which is associated with the
closed quadratic form
Z π2 2
2
|∇u| − 2 |u|
Qf [u] =
dx
(10)
d
Ωf
with the form domain H01 (Ωf ). From the assumptions on f it follows that
σess (Af ) = [0, ∞) .
Let us denote by −µj the non decreasing sequence of negative eigenvalues of Af taking into account their multiplicities. We shall estimate the
total number of −µj by the number of negative eigenvalues of a certain
two-dimensional Schrödinger operator −∆ − Vf with Vf depending on the
deformation function f .
4
Theorem 2. Assume that the function f : R2 → R is in C 2 (R2 ) and such
that suppf ⊂ B(R) for some R > 0, and kf k∞ < d. For any t ≥ 0 we have
NL2 (Ωf ) (Af − t) ≤ NL2 (R2 ) (−∆ + 3Vf − 3t) ,
(11)
where
Vf =
π2
π2
−
− b1 |∇f |2 − b2 (R) |∆f |2 − b3 (R) |∇f |4 ,
(d + f )2
d2
with b1 , b2 (R) and b3 (R) satisfying (19).
Proof. We write a given trial function ψ ∈ H01 (Ωf ) as
ψ(x, y, z) = ϕ(x, y, z) g(x, y) + h(x, y, z) ,
(12)
where
ϕ(x, y, z) =
and
Z
0
Hence
Z Ωf
s
d+f (x,y)
2
sin
d + f (x, y)
πz
d + f (x, y)
g ∈ H 1 (R2 )
ϕ(x, y, z)h(x, y, z) dz = 0 ∀ (x, y) ∈ R2 .
π2
|∇ψ| − 2 |ψ|2
d
2
dx dy dz =
Z
Ωf
(13)
|∇ϕ|2 |g|2 + |∇x,y g|2 |ϕ|2 + |∇h|2
π2
− 2 (|ϕ g|2 + |h|2 ) + 2ggx′ ϕ′x ϕ + 2gϕ′y h′y + 2ggy′ ϕ′y ϕ + 2gϕ′x h′x
d
+ 2gϕ′z h′z + 2ϕgx′ h′x + 2ϕgy′ h′y dx dy dz .
(14)
Here and in the sequel we will use the shorthand u′x = ∂u
∂x and analogously
for other partial derivatives. We estimate all the mixed terms in (14), except
for the last three, point-wise in the following way:
′ 2
′ 2
2g gx′ ϕ′x ϕ ≤ a−1
1 |ϕ gx | + a1 |gϕx | ,
′ 2
′ 2
2g gy′ ϕ′y ϕ ≤ a−1
1 |ϕ gy | + a1 |gϕy | ,
′ 2
′ 2
2gϕ′x h′x ≤ a−1
2 |hx | + a2 |gϕx | ,
′ 2
′ 2
2g ϕ′y h′y ≤ a−1
2 |hy | + a2 |gϕy | ,
5
(15)
where a1 and a2 are real positive numbers whose values will be specified
later. Furthermore, from integration by parts and (13) follows that
Z
Z
gϕ′′z h dxdydz = 0 .
gϕ′z h′z dxdydz = −
Ωf
Ωf
integrating by parts again and using (13) we can rewrite the last two terms
in (14) as
Z
Z
Z
′
′
′ ′
g(ϕ′′x h + ϕ′x h′x ) dxdydz ,
ϕx h gx dxdydz =
ϕ hx gx dxdydz = −
Ωf
Ωf
Ωf
Ωf
Z
ϕ h′y gy′ dxdydz = −
Z
Ωf
ϕ′y h gy′ dxdydz =
Z
Ωf
g(ϕ′′y h + ϕ′y h′y ) dxdydz .
The terms 2gϕ′x h′x and 2gϕ′y h′y will be estimated in the same way as in (15).
For the rest we use the following point-wise inequalities
2
2g ϕ′′x h ≤ a3 g2 |ϕ′′x |2 + a−1
3 h χf ,
2
2g ϕ′′y h ≤ a3 g2 |ϕ′′y |2 + a−1
3 h χf ,
where χf denotes the characteristic function of the support of f . Now we
put a1 = a2 = 3 and arrive at
Z Z π2
1
2
2
2
2
|∇ψ| − 2 |ψ|
|∇x,y g| + Ṽf (x, y)|g|
dx dy dz ≥
dx dy
d
3
Ωf
R2
Z π2
1
2
|∇x,y h|2 + |h′z |2 − 2 |h|2 − a−1
|h|
χ
dx dy dz
(16)
+
f
3
3
d
Ωf
with
Ṽf =
π2
π2
− 2 −
2
(d + f )
d
Z
0
d+f
8 |ϕ′x |2 + |ϕ′y |2 + a3 |ϕ′′x |2 + |ϕ′′y |2 dz .
Since h satisfies Dirichlet boundary conditions at ∂Ωf and f < d, we deduce
from (13) that
Z π2 2
1
−1
2
′ 2
2
|∇x,y h| + |hz | − 2 |h| − a3 |h| χf dx dy dz
3
d
Ωf
Z 1
4π 2
π2
2
2
−1
2
≥
|∇x,y h| +
− 2 |h| − a3 |h| χf dx dy dz
3
(d + f )2
d
Ωf
Z dZ 3π 2
3π 2
1
2
|∇x,y h|2 + 2 |h|2 − a−1
+
|h|
χ
dx dy dz
≥
f
3
3
d
d2
0
R2
Z 2d Z
1
−1
2
2
+
|∇x,y h| − a3 |h|
dx dy dz
(17)
3
d
suppf
6
From the fact that the support of f is compact it follows that the last term in
(17) is non-negative for all a3 ≥ λ−1 (R), where λ(R) is the lowest eigenvalue
of −∆x,y on the disc B(R) with Dirichlet boundary conditions. Moreover,
the expression on the third line of (17) can be bounded from below as follows
dZ
1
3π 2 2
3π 2
−1
2
2
|∇x,y h| + 2 |h| − a3 + 2
|h| χf dx dy dz (18)
3
d
d
0
R2
Z d Z 3π 2 2
1
−1
2
2
|∇r,θ h| + 2 |h| χ[R,∞) − a3 |h| χ[0,R] rdr dθ dz ,
≥
3
d
0
R2
Z
where we have used the polar coordinates (r, θ)nin R2 . Inoview of Lemma
d2
1, see Appendix, (18) is positive for a3 ≥ max 8 R2 , 3π
2 . Therefore we
choose
2
d
2
−1
, 8 R , λ (R) .
a3 (R) = max
3π 2
Now it remains to estimate the first term on the right hand side of (16). By
a direct calculation we arrive at
Z d+f
5 |ϕ′x |2 + |ϕ′y |2 + a3 (R) |ϕ′′x |2 + |ϕ′′y |2 dz ≤ b1 |∇f |2
0
+ b2 (R) |∆f |2 + b3 (R) |∇f |4
where b1 , b2 (R), b3 (R) are positive numbers which satisfy
2
4π 2
π
a3 (R) π 2
π4
b1 ≤
, b2 (R) ≤
, b3 (R) ≤ 4 a3 (R)
+
.
5 d2
d2
d4
5d2
Finally, combining (16) and (13) we obtain
Z π2
2
2
2
dx dy dz
|∇ψ| − 2 |ψ| − t|ψ|
d
Ωf
Z
1
≥
|∇x,y g|2 + 3Vf (x, y)|g|2 − 3t|g|2 dx dy ,
3 R2
(19)
(20)
holds true for any t ≤ 0.
Let us show that (20) implies (11). We introduce the subspace Mt ⊂ L2 (R2 )
spanned by the eigenvectors associated with the negative eigenvalues of the
operator
1
(−∆ + 3Vf − 3τ )
in L2 (R2 ),
3
7
and define Mt ⊂ L2 (Ωf ) by
Mt = {g ϕ : g ∈ Mt } .
Obviously
dim Mt ≤ NL2 (R2 )
1
(−∆ + 3Vf − 3τ )
3
= NL2 (R2 ) (−∆ + 3Vf − 3t) .
that ψ ⊥ Mt and write ψ = g̃ ϕ + h. Then g̃ ϕ ⊥ Mt and since
RAssume
d+f
|ϕ(x,
y, z)|2 dz = 1 for all (x, y) ∈ R2 , this means that g̃ ⊥ Mt . In view
0
of (20) this implies
Z π2
2
2
2
dx dy dz ≥ 0 .
|∇ψ| − 2 |ψ| − t|ψ|
d
Ωf
By the variational principle we conclude that
NL2 (Ωf ) (Af − t) ≤ dim Mt = dim Mt ≤ NL2 (R2 ) (−∆ + 3Vf − 3t) .
Remark 1. From the assumption f < d it follows that all negative eigenvalues of Af come from the first channel only. However, we would like to
mention that this assumption is purely technical and could be replaced by
f < nd, n ∈ N. In that case we would have to use another decomposition of
a test function ψ, analogous to (12), taking into account also the functions
associated with higher transversal modes in z. For the sake of simplicity we
therefore suppose f < d.
Corollary 1. For any p > 1 there exist positive constants C1 and Cp such
that
X
p
+ Cp kVf kL1 (R+ ,Lp (S1 )) (21)
Fs (µj ) ≤ C1 Vf ln( x2 + y 2 /s) 1
L (B(s))
j
holds for all s > 0.
Proof. Since Fs′ is non-negative we have
Z ∞
X
Fs′ (t) NL2 (Ωf ) (Af − t) dt
Fs (µj ) =
0
j
≤
Z
≤ 3
∞
0
Z
Fs′ (t) NL2 (R2 ) (−∆ + 3Vf − 3t) dt
∞
0
Fs′ (t) NL2 (R2 ) (−∆ + 3Vf − t) dt = 3
and the statement follows from Theorem 1.
8
X
j
Fs (λj ) .
The disadvantage of estimate (21) is the presence of the terms in Vf which
contain the derivatives of f . Firstly, small oscillations of f will lead to the
unnecessary growth of the right hand side in (21). Secondly, the deformation
function f in general need not be C 2 −smooth. This can remedied using the
monotonicity property of eigenvalues of Laplace operators in domains with
Dirichlet boundary conditions. Namely, for any f˜ ≥ f we have
NL2 (Ωf ) (Af − t) ≤ NL2 (Ωf˜) (Af˜ − t)
∀t ≥ 0.
As an immediate consequence of Theorem 1 and Corollary 1 we thus get
Theorem 3. Let 0 ≤ f < d be a continuous function with support in B(R).
Then there exist constants C3 and C4 such that
X
r
Fs (µj ) ≤ inf C3 Vf˜ ln + C4 kVf˜kL1 (R2 ) ,
(22)
s L1 (B(s))
f˜≥f
j
where the infimum is taken over all radially symmetric functions f˜ ∈ C02 (B(R)).
Remark 2. Let us consider the behaviour of the estimate (21) for weakly
deformed layers. This means replacing f by α f and letting α go to zero.
Theorem 2 and the result of [10] yield the following upper bound on the
number of negative eigenvalues of Aαf :
r
+ kVαf˜kL1 (R2 ) .
NL2 (Ωf ) (Aαf ) ≤ 1 + const Vαf˜ ln s L1 (R2 )
From the explicit form of Vαf˜ thus follows that Aαf has only one negative
eigenvalue, −µ1 (α), for α small enough. Moreover, inequality (21) implies
C(f, d)
|µ1 (α)| ≤ exp −
,
(23)
w(α)
where C(f, d) is a positive factor independent of α and
w(α) = α + O(α2 )
α → 0.
(24)
This agrees, in order of α, with the asymptotics found in [1].
Appendix
Lemma 1. Let u ∈ H 1 (R+ , r dr). Then for any R > 0 the inequality
Z
Z 2R
Z R
8 2 2R ′ 2
2
2
|u | r dr
(25)
|u| r dr + R
|u| r dr ≤
3
0
R
0
holds true.
9
Proof. Let us define the function h : R+ → R by

0<r≤R
 1
h(r) =
R < r < 2R .
1 − r−R
R

0
2R ≤ r
For any r ∈ (0, R) we then have
Z
2R
(hu)′ (t) dt
u(r) = h(r)u(r) = −
r
Z 2R
Z 2R
1
hu′ dt .
u dt −
=
R R
r
(26)
The Cauchy-Schwarz inequality thus implies
Z 2R
Z 2R
2
|u(r)|2 ≤
|u|2 dt + 2khk2
|u′ |2 dt .
R R
r
Multiplying by r and integrating over (0, R) we get
Z
R
0
2
|u| r dr ≤
Z
2R
R
2
2
|u| r dr + 2R khk
To conclude the proof we note that khk2 =
4
3
Z
0
2R
|u′ |2 r dr .
R.
Acknowledgement
The support from the DFG grant WE 1964/2 is gratefully acknowledged.
The authors are grateful to the referee for important corrections of the original text.
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