Download Killip-Simon problem on two disjoint intervals

Killip-Simon problem on two disjoint intervals
Benjamin Eichinger
Institute of Analysis
Johannes Kepler University Linz
July 20, 2015
B. Eichinger
Killip-Simon problem on two disjoint intervals
Killip-Simon theorem
Theorem
Let dσ(x) = σ 0 (x)dx + dσsing (x) be the spectral measure of a
one-sided Jacobi matrix J+ . Then the following are equivalent:
◦
(op)
J+ − J+ ∈ HS
(sp) The spectral measure dσ is supported on X ∪ [−2, 2] and
Z 2
p
3
Xq
| log σ 0 (x)| 4 − x 2 dx +
xk2 − 4 < ∞.
−2
xk ∈X
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Killip-Simon problem on two disjoint intervals
Isospectral torus
S
Let E = [b0 , a0 ] \ gj=1 (aj , bj ) and for a two-sided Jacobi matrix J
we define
E
D
r± (z; J) = (J± − z)−1 e −1±1 , e −1±1 .
2
2
Definition
A Jacobi matrix J is called reflectionless on E if
a02 r+ (x + i0) =
1
for almost all x ∈ E .
r− (x − i0)
The class J(E ) is formed by Jacobi matrices, which are
reflectionless on their spectral set E .
B. Eichinger
Killip-Simon problem on two disjoint intervals
Spectral theory of periodic Jacobi matrices
Theorem
Let E = [b0 , a0 ] \
Sg
j=1 (aj , bj ).
Then E is the spectrum of a p
◦
periodic two-sided Jacobi matrix J if and only of there exists a
polynomial Tp (z) such that Tp−1 ([−2, 2]) = E with some additional
properties.
Theorem (Magic formula)
S
Let E = [b0 , a0 ] \ gj=1 (aj , bj ) be the spectrum of a p periodic
◦
Jacobi matrix. Then J ∈ J(E ) if and only if
◦
Tp (J) = S −p + S p .
B. Eichinger
Killip-Simon problem on two disjoint intervals
Damanik-Killip-Simon theorem
Theorem (Damanik-Killip-Simon theorem)
The following are equivalent:
p
∗ )p ) ∈ HS
T (J+ ) − (S+
+ (S+
(op)
(sp) The spectral measure dσ is supported on X ∪ E and
Z
Xp
p
3
dist(xk , E ) < ∞.
| log σ 0 (x)| dist(x, R \ E )dx +
E
xk ∈X
Why does this Magic Formula only exist in the periodic
case?
What can we do in the general case?
B. Eichinger
Killip-Simon problem on two disjoint intervals
The Riemann surface RE
From now on, we assume that E = [b0 , a0 ] \ (a1 , b1 ).
We define
ℛ+
s(z) = (z−a0 )(z−b0 )(z−a1 )(z−b1 )
b0
⊕
⊖
a1
b1
⊕
⊖
a0
a1
b1
⊖
⊕
a0
and
RE = {(P = (z, w ) : w 2 = s(z)}∪{∞+ , ∞− }
the compact Riemann surface of
s(z).
It is convenient to imagine RE
as a two-sheeted
p Riemann surs(z) > 0, for
face, where
z > a0 on R+ .
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ℛ-
b0
⊖
⊕
Killip-Simon problem on two disjoint intervals
The isospectral torus
By the reflectionless property we extend r+ to a meromorphic
function on RE :
(
a02 r+ (z) if P = (z, w ) ∈ R+
a02 r+ (P) =
1
if P = (z, w ) ∈ R− .
r− (z)
Let us define
DE = {(x, ) : x ∈ [a1 , b1 ], = ±1}.
One can show that for any J ∈ J(E ) we have
1 w + w0
a02 r+ ((z, w )) = (
− (z − q0 )),
2 z − x0
where (x0 , 0 ) ∈ DE , q0 = q0 ((x0 , 0 )) ∈ R and w0 = w0 (x0 , 0 ).
Moreover, each such a function defines J ∈ J(E ).
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Killip-Simon problem on two disjoint intervals
The isospectral torus, Abel map
J(E ) o
/ DE
>
r
a
!
A
~
T
S+
(x0,1)
b0
a1
(x0,-1)
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b1
a0
Killip-Simon problem on two disjoint intervals
The Abel map
There exists a conformal map from RE onto a rectangle with
1−τ 1+τ −1+τ
vertices −1−τ
in C. It is given by
2 , 2 , 2 , 2
Zz
u(z) = C
a0
dz
p
,
s(z)
τ/2
uc
S+
u0
x0
-1/2
u∞ -
u∞ +
1/2
S-
x1
-τ/2
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Killip-Simon problem on two disjoint intervals
The Abel map
We call the map
u∞-
u∞ +
A : RE → C/L
z 7→ u(z),
u∞-
u∞ +
where L = Z + τ Z, the Abel map.
B. Eichinger
Killip-Simon problem on two disjoint intervals
Meromorphic functions on RE and elliptic functions
Definition
Meromorphic functions on C with periods 1 and τ bear the name
elliptic functions.
Theorem
The properly counted (i.e., counted with multiplicity taking into
account) number of poles of a nonconstant elliptic function f (u) in
a period parallelogram is equal to the properly counted number of
zeros.
Theorem
N
Let {αk }N
k=1 , {βk }k=1 are a system of points in a period
parallelogram. Then αk are the zeros on βk are the poles of an
N
N
P
P
elliptic function, if and only if
αk −
βk ∈ L.
k=1
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k=1
Killip-Simon problem on two disjoint intervals
Meromorphic functions on RE and elliptic functions
one-to-one correspondence
between meromorphic functions
on RE and elliptic functions
w.r.t. L.
τ/2
u1
-1/2
uc
u∞-
u0
u∞ +
1/2
r+ , meromorphic function on
RE with poles at {∞− , x0 } and
zeros at {∞+ , x1 }.
u∞+ + u1 − (u∞− + u0 ) ∈ Z
-τ/2
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Killip-Simon problem on two disjoint intervals
Shift on J(E ) and on T
One can show that
r+ (z, w ) = −
1
(1)
.
b0 − z + r+ (z, w )
(1)
Thus, the zero of r+ is the pole of r+ .
Let µ = u∞+ − u∞− . Then u1 = u0 − µ mod Z.
μ
r+
J(E ) o
/ DE
>
a
!
~
u1
u0
A
T
B. Eichinger
Killip-Simon problem on two disjoint intervals
Functional models for Jacobi matrices
Theorem (Jacobi theta function)
There exists an analytic function θ1 on C with the following
properties:
θ1 (u ± 1) = −θ1 (u)
θ1 (u ± τ ) = −e −iπτ e ∓2πiu θ1 (u)
Moreover, the zeros of θ1 are given by
v = m + nτ, m, n ∈ Z.
Recall, that r+ is a meromorphic functions on RE with poles at
{∞− , x0 } and zeros at {∞+ , x1 }. Thus, it is given by
r+ (z, w ) = C
θ1 (u − u∞+ )θ1 (u − u1 )
.
θ1 (u − u∞− )θ1 (u − u0 )
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Killip-Simon problem on two disjoint intervals
Functional models for Jacobi matrices
Now we set
B(u) =
and
k α (u) = C
θ1 (u − u∞+ )
θ1 (u − u∞− )
θ1 (u − u0 )
,
θ1 (u − uc− )
k α (u)
K α (u) = p
,
k α (u∞+ )
where c− = x− + i κ2 is a certain normalization point.
This functions are no elliptic functions! We have
B(u + 1) = B(u) and K α (u + 1) = K α (u),
and
B(u + τ ) =
θ1 (u − u∞+ + τ )
θ1 (u − u∞+ ) e −i(u−u∞+ )
=
.
θ1 (u − u∞− + τ )
θ1 (u − u∞− ) e −i(u−u∞− )
Thus, we see that
B(u + τ ) = e iµ B(u).
B. Eichinger
Killip-Simon problem on two disjoint intervals
Functional models for Jacobi matrices
Similarly, we can show that
K α (u + τ ) = e iα K α (u),
where α = u0 − uuc− . In general, we define
K α−kµ (u) =
θ1 (u − (u0 − kµ))
.
θ1 (u − uc− )
Again, we see that
K α−kµ (u + τ ) = e i(α−kµ) K α−kµ (u).
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Killip-Simon problem on two disjoint intervals
The space H 2 (α)
Let Ω = C \ E . For α ∈ T, we define H 2 (α) as the space of
multivalued, analytic functions f α (z) in Ω, which satisfy:
|f α |2 has a harmonic majorant in Ω,
g α (u) = f α (z), satisfies g α (u + τ ) = e iα g α (u).
Note, that |f α |2 is single-valued. The norm is given by the value at
infinity of the least harmonic majorant of |f α |2 . The corresponding
scalar product is given by
Z
α α
hf1 , f2 i =
f1α (x)f2α (x)ω(dx, ∞),
E
where ω(dx, ∞) denotes the harmonic measure of infinity of the
domain Ω.
It is given by
i x −c
p
ω(dx, ∞) =
.
2π s(z)
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Killip-Simon problem on two disjoint intervals
Functional models for Jacobi matrices
Theorem
Let
enα (z) = B n (z)K α−kµ (z)
Then
en n ∈ N is a ONB of H 2 (α)
en n ∈ Z is a ONB of L2 (α)
Theorem
Let enα be defined as above, then the multiplication operator z in
L2 (α) with respect to the basis {enα } is a Jacobi matrix J = J(α):
α
α
zenα = an (α)en−1
+ bn (α)enα + an+1 (α)en+1
,
B. Eichinger
Killip-Simon problem on two disjoint intervals
Sketch of the proof
We only sketch, why the multiplication by z is a Jacobi matrix. We
have
r+ (z) = −
θ1 (u − u∞+ )θ1 (u − u1 )
1
.
=C
2
(1)
θ1 (u − u∞− )θ1 (u − u0 )
z − b0 + a1 r (z)
That is,
a0 r+ = −B
K α−µ
Kα
(1)
and a1 r+ = −B
K α−2µ
.
K α−µ
Thus we have
zBK α−µ
BK α−2µ
a0 K α
=
(z
−
b
)
−
a
⇔
0
1
BK α−µ
K α−µ
= a0 K α + b0 BK α−µ + a1 B 2 K α−2µ ⇔
ze1α = a0 e0α + b0 e1α + a1 e2α
B. Eichinger
Killip-Simon problem on two disjoint intervals
Proof of the magic formula
S −2 JS = J ⇔ 2µ ∈ Z. This is only a property of the spectrum not
of J! Consider, in this case the basis {enα } has the following form
{. . . , B −1 K α+µ , K α , BK α+µ , B 2 K α , . . . }.
Recall µ = u∞+ − u∞− . There exists a function
θ1 (u − u∞+ ) 2
ψ(z, w ) =
= B(u)2 ,
θ1 (u − u∞− )
on RE .
ψ + ψ −1 is a function on Ω = C \ E . Thus,
ψ(z, w ) +
1
= T2 (z).
ψ(z, w )
This is the magic formula.
`2
L2 (α)
en
enα
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S2
ψ
J(α)
z
Killip-Simon problem on two disjoint intervals
Functional models in the general case
Now let us consider the case 2µ ∈
/ Z. In this case we define
ψ=
θ1 (u − u∞+ )θ1 (u − uc+ )
= B(u)Bc (u),
θ1 (u − u∞− )θ1 (u − uc− )
where µ + µc ∈ Z. In this case we have
ψ(z, w ) +
1
λ1
= λ 0 z + d0 +
= ∆(z).
ψ(z, w )
d1 − z
B. Eichinger
Killip-Simon problem on two disjoint intervals
Functional models in the general case
Let k α (z, c) = kcα (z) be the reproducing kernels of the space
H 2 (α). We define Kψ (α) = H 2 (α) ψH 2 (α). Kψ is spanned by:
f0α (z) = λ(α)
kdα1 (z)
kkdα1 (z)k
and f1α (z) = Bd1 (z)
k α+µ (z)
,
kk αµ k
Thus, H 2 (α) = Kψ (α) ⊕ ψKψ (α) + ψ 2 Kψ (α) ⊕ . . . .
Theorem
The system of functions
fnα
(
ψ m f0α , n = 2m
=
ψ m f1α , n = 2m + 1
forms an orthonormal basis in H 2 (α) for n ∈ N and
forms an orthonormal basis in L2 (α) for n ∈ Z.
B. Eichinger
Killip-Simon problem on two disjoint intervals
Functional models in the general case, SMP-matrices
Definition
We call the operator A(α) ∈ L(`2 ), which corresponds to the
multiplication by z in this basis a SMP-matrix.
A(α) is two periodic.

..
σ(A(α)) = E

.
 .
 .. ... A


A∗ B A

A=
A∗ B



A∗

A
..
.
..
.
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

 A= 0 0 ,

p0 p1

,

p0 q0 + d0 p1 q0

.
..  B =
p1 q0
p1 q1
. 
Killip-Simon problem on two disjoint intervals
By construction, we obtain the magic formula for SMP matrices.
`2
L2 (α)
en
fnα
S2
ψ
A(α)
z
Theorem
Let A(E ) be the set of all SMP matrices of period two with their
spectrum on E. Then we have:
A ∈ A(E )
⇔
∆(A) = λ0 A + d0 + λ1 (A − d1 )−1 = S 2 + S −2
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Killip-Simon problem on two disjoint intervals
The Jacobi flow on SMP-matrices
Definition
We define the map F : A(E ) → J(E ), by F(A(α)) = J(α).
Lemma
Let kP+ A(α)e−1 k2 and {ak , bk } be the Jacobi parameters of J(α).
Then we have
p1 q1 = b−1 , p02 + q02 = a02
α =fα .
It follows immediately, since e−1
−1
B. Eichinger
Killip-Simon problem on two disjoint intervals
The Jacobi flow on SMP-matrices
Definition
The Jacobi flow in A(E ) is defined by J (A(α)) = A(α − µ).
A
F
/J
J
A
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S
F
/J
Killip-Simon problem on two disjoint intervals
Explicit formulas for the Jacobi flow
Theorem
Let U(α) be the periodic 2 × 2-block diagonal unitary matrix given
by
(1)
U(α) e2m e2m+1 = e2m e2m+1 u(α),
where
1
u(α) = q
p02 (α) + p12 (α)
p0 (α) p1 (α)
p1 (α) −p0 (α)
(2)
Then
J A(α) := A(µ−1 α) = S −1 U(α)∗ A(α)U(α)S.
B. Eichinger
(3)
Killip-Simon problem on two disjoint intervals