Download hilbert theorem on lemniscate and the spectrum of the perturbed shift

HILBERT THEOREM ON LEMNISCATE AND
THE SPECTRUM
OF THE PERTURBED SHIFT
1
2
V.L.Oleinik , B.S.Pavlov
1
Department of Physics, St.Petersburg State University, Ulianovskaya 1, Petrodvorets,
St.Petersburg, 198904, Russia
2
Department of Mathematics, University of Auckland, Private Bag 92019,
Auckland, New Zealand
Department of Mathematics report series 4....., May 1999
The University of Auckland, New Zealand.
Abstract.
The spectrum of the perturbed shift operator T : f (n) ! f (n +1)+ a(n)f (n) in l2(Z)
is considered for periodic a(n) and xed constant > 0. It is proven that the spectrum
is continuous and lls a lemniscate. Some isospectral deformations of the sequence a(n)
are described. Similar facts for the perturbed shift in the spaces of sequences of some
hypercomplex numbers is derived.
1 Introduction
1.1 Lemniscate
Following 8], by a lemniscate we mean the locus of a point z 2 C for which the product
of distances to a nite number of xed points fz1 z2 ::: zN g 2 C N is equal to the xed
positive constant rN . The condition may be written in the form
j(z ; z1 )(z ; z2 ):::(z ; zN )j = rN > 0:
(1)
The lemniscate with equation jz2 ; 1j = r2 is called the Cassini oval, or, in the case r = 1,
the Bernoulli lemniscate ( which looks like the \innity"-sign 1).
Some properties of the lemniscate (1) can be deduced from the maximum modulus
principle for an analytic function:
1) the lemniscate (1) separetes each point zk from innity
2) no point of the lemniscate (1) can lie interior to Jordan curve consisting wholly of the
points of (1) 3) each such Jordan curve must contain inside at least one point zk 4) therefore the lemniscate (1) consists of a nite number of bounded Jordan curves.
For other basic properties of lemniscates the reader is referred to 5], 8], 9].
1.2 Hilbert theorem on lemniscate
For the larger number N , we may obtain lemniscates (1) of most varied appearance. In
fact one can approximate the boundary of any bounded simply connected domain by a
lemniscate (see 3], 5] p. 379).
Theorem (D. Hilbert, 1897) For any bounded simply connected domain D and for
any " > 0 there exists a lemniscate l D which consists of the single continuum1 such
that the boundary @D of the domain D lies in the "-neighborhood of l and the lemniscate
l lies in the "-neighborhood of the boundary @D, i.e.
@D fz 2 C : dist (z l) < "g
l fz 2 C : dist (z @D) < "g:
One can nd in 8], 9]. the extention of Hilbert's theorem to the general case of an
unbounded multiply-connected domains with a compact complement.
1.3 Shift operator
We denote by S the shift operator on the doubly innite sequences of the complex numbers, i.e.
(Sf )(n) = f (n + 1) n 2 Z:
1
This important condition is omitted by the author of the book 5].
2
The operator S is a unitary operator on l2 (Z) and the unit circle is its continuous spectrum.
We consider the scaled operator S with a positive number , and perturbed by adding
a multiplication operator by some function a : Z ! C . So, we consider the operator
T = S + a. One can see that the description of the spectrum (T ) of T in l2(Z) may be
reduced to nding all complex numbers z such that the homogeneous dierence equation
f (n + 1) + a(n)f (n) = zf (n) n 2 Z
has a nontrivial solution2 and each solution is bounded. We denote the set of all such
numbers z by s(T ) (T ) = s(T ) .
Our nearest aim is to show that for every > 0 and for every periodic function a
the set s(T ) is a lemniscate and each lemniscate (1) coincides with the set s(T ) for some
periodic function a with a period N0 N . On the other hand, due to Hilbert's theorem,
for every set ; which is the boundary of a simply connected domain D (i.e. ; = @D)
there exist a periodic function a and a positive number such that the analytic curve
s(T ) approximates ;. In fact using the extention of Hilbert's theorem to the case of
unbounded multiply connected domains with a bounded complement (see 6], 7], 8], 9])
we can approximate the boundary of every closed bounded set with connected complement
by a lemnicsate and therefore by the set s(T ) for some periodical function a.
The main theorem is formulated at the end of section 2 and it is proved in the section
3. In the section 4 we consider the perturbed shift on sequences of quaternions and Cayley
numbers.
2 Main Theorem
Let f be a two-sided sequence of the complex numbers, i.e. f :
supn jf (n)j. By S we denote the shift operator
Z
! C . Put jf j :=
(Sf )(n) = f (n + 1) n 2 Z:
The shift S is a unitary operator on
l2(Z) := ff :
X
n2Z
jf (n)j2 < 1g:
It means that kSf kl2(Z) = kf kl2(Z) for each f 2 l2 (Z).
For a given sequence a : Z ! C and a positive number > 0 we consider the perturbed
shift
Ta := S + a:
Note that the operator Ta is bounded on l2(Z) if and only if the function a(n) is bounded,
i.e. jaj < 1.
2 It means that inf fn : a(n) = zg > ;1.
3
We say that a complex number z belongs to the spectrum s(Ta ) of the operator Ta
if and only if the following dierence equation
Sf + af = zf
f (n + 1) + a(n)f (n) = zf (n) n 2 Z
(2)
has a nontrivial solution and every solution f is bounded, i.e. jf j < 1.
If we put f (0) = 1 then for every z 2= fa(n) : n < 0g the corresponding solution of the
equation (2) has the following explicit form
8 ;n Qn; (z ; a(k)) n 1
<
k
n = 0
f (n) = : Q 1 j
n
j
j
n
j
;
k (z ; a(;k)) n ;1:
1
=0
1
=1
(3)
On the other hand, if z = a(n0 ) then for every solution f of the equation (2) we get
f (n) 0, n > n0 .
Remarks. 1. For every > 0 the set s(Ta ) is a translation-invariant one, that is
for every m 2 Z we have s(Ta ) = s(Ta ) with a0 (n) = a(n + m) n 2 Z. Really, if
Ta f = zf then Ta f 0 = zf 0 with f 0(n) = f (n + m). Therefore we can suppose that
a(n) 2= s(Ta ) for n < 0 (see the footnote2).
2. The spectrum s(Ta ) coincides with the set of the stability of the equation (2) (see 1],
theorem 5.4.1).
3. Let a(n) be a stationary sequence, a0 := a(0) = a(n) n 2 Z, then the set s(Ta ) is a
circle of radius centered at the point a0 . Indeed, let f be a solution of the equation (2)
then by (3) f (n) = f (0)(z ; a0)n=n n 2 Z, is a geometric progression and the solution
f is bounded if and only if jz ; a0 j= = 1. In particular s(S ) = fz 2 C jzj = 1g is the
unit circle.
For a xed positive integer N let us consider a nite ordered set of complex numbers
~a = fa1 a2 ::: aN g 2 C N and a sequence of permutations n of the set N~ = f1 2 ::: N g,
n 2 S (N~ ) n 2 Z, i.e. : Z ! S (N~ ) (see 4], p.77). For given ~a and we construct
a function a~a : Z ! C such that for every n = kN + m with k 2 Z and m in the set
f0 1 ::: N ; 1g
a~a(n) = a~a(kN + m) = ak (m+1) :
(4)
Note that for a stationary sequece = f0 g : n = 0 n 2 Z, we get the periodic
function a(n) = a~a(n) with the period N0 N , i.e. a(n + N0 ) = a(n), n 2 Z. For
instance, if all numbers of the set ~a are dierent from each other then N0 = N for every
0 2 S (N~ ). Denoting by e the unit element of the permutation group S (N~ ) and choosing
= feg : n = e n 2 Z, we have got the following periodic sequence a~a := afeg~a , that
is
~ k 2 Z:
a~a (n + kN ; 1) = an n 2 N
For given > 0 and a set ~a let us consider the polynomial
0
0
P~a (z) := (z ; a1 )(z ; a2 ):::(z ; aN ):
4
The locus of a point for which the product of distances from points ~a is the positive
constant N is a lemniscate l~a. Thus, the lemniscate l~a is dened by the equation
jP~a(z)j = N :
We now proceed to formulate our main
Theorem. For each xed positive number ,each set ~a of complex numbers, and every
sequence = fngn2Z of permutations n of the set N~ = f1 2 ::: N g we have
s(Ta ) = s(Ta ) = l~a 0
where a = a~a and a0 = a~a .
3 Proof of the Main Theorem
Proof. Fix a permutation 0 2 S (N~ ) and put
~a0 := fa0(1) a0(2) ::: a0(N ) g:
Then from the equation
P~a0 (z) P~a (z) = (z ; a1 )(z ; a2 ):::(z ; aN )
(5)
we see that the lemniscates generated by ~a and ~a0 coincide for every > 0, i.e.
l~a = l~a0 :
The set l~a is bounded and no point of the set ~a can lie on the lemniscate l~a.
Therefore there are two positive constants d = d(~a ) and D = D(~a ) such that
d jz ; anj D
(6)
for every z 2 l~a and n 2 N~ .
Fix , ~a, and z. Let f be a solution of the equation (2) with a = a~a . Then from
the last equation, the denition (4) of the function a, and the property (5) we get
f (N ) = 1 (z ; a~a(N ; 1))f (N ; 1) = 1 (z ; a0(N ) )f (N ; 1) =
1 (z ; a )(z ; a
1
(7)
0 (N )
0 (N ;1) )f (N ; 2) = ::: = N P~a (z )f (0):
2
We let p(z) := ;N P~a(z), and for every positive integer k using the equation (7) we
obtain f (kN ) = p(z)k f (0). Hence the condition z 2 s(T~a) implies supk>0 jf (kN )j < 1,
or jp(z)j 1. On the other hand by the equation (2) we have (z 2= ~a)
f (;1) = z ; a(;1) f (0) = z ; a
f (0)
1 (N )
;
5
and nally
f (;2) = (z ; a
2
1 (N ;1) )(z ; a
;
;
N))
1(
f (0)
f (;N ) = fp((0)
z) :
Therefore for each positive integer k we obtain
f (;kN ) = pf((0)
z)k :
(8)
So, z 2 s(Ta ) implies z 2= ~a and jp(z)j 1. At last we have
jp(z)j = 1:
(9)
It means that
s(Ta) l~a = fz 2 C : jp(z)j = 1g:
To complete the proof of the theorem note that for each z 2 l~a and n = kN + m
with m in f1 ::: N ; 1g, k 2 Z, using (6), (9) we have the following estimates
jf (n)j (D=)mjf (0)j (cD)N ;1jf (0)j if k 0
and
jf (n)j (=d)mjf (0)j (c=d)N ;1jf (0)j if k < 0
with c := maxf 1=g. So, the condition z 2 l~a implies z 2 s(T~a), and
s(Ta ) = l~a = s(Ta ):
End of the proof.
Remarks. 1. Every lemniscate (1) is the spectrum of some operator Tra . For
example, we can choose a = afeg~a with ~a = fz1 z2 ::: zN g. Then a(n) is a periodic
function. Therefore
2. every lemniscate (1) is the spectrum of some operator Tra with a periodic function
a(n).
3. For every lemniscate (1) such that z1 6= z2 and for every positive integer k there exists
a periodical function a(n) with the period kN such that the spectrum s(Tra ) coincides
with the lemniscate (1). For example, we can construct the desired function as a = afeg~a
with the ordered set ~a = fa1 a2 ::: akN g which consists of the (k ; 1) identical blocks:
z1 z2 ::: zN ], but the last one is z2 z1 ::: zN ].
4. Let a = a~a then for every nite function b : Z ! C , i.e. b(n) 0 for n : n(n ; N0 ) >
0, we have
s(Ta ) = s(Ta+b )
and no point of the complex plain C is an eigenvalue of the operator Ta+b on the space
l2 (Z). The latter statement means that the corresponding solution f of the equation
(S + a + b)f = zf
0
6
belongs to l2 (Z) if and only if f 0.
Really, the embedding s(Ta+b) la is obviuos. The polynomial
(z ; a(0) ; b(0))(z ; a(1) ; b(1)):::(z ; a(N0) ; b(N0 ))
is bounded on the lemniscate la , hence z 2 la implies z 2 s(Ta+b ). On the other hand
any l2 (Z)-solution f is bounded, jf j < 1. Therefore by (8), (9) we get jf (;kN )j = jf (0)j
for every positive integer k. So, f (0) = 0 = f (n), n 2 Z.
5. Let a = a~a then by the same way as above we can prove that for every function
b : f0 1 2 ::: N0g ! C we have
s(Ta ) = s(Ta )
0
with
8 a(n)
<
b(n)
a0 (n) = :
a(n ; N ; 1)
if n < 0
if 0 n N0 if n > N0:
0
No point of the complex plain C is an eigenvalue of the operator Ta on the space l2 (Z).
Example. Let m0, m1 be nonnegative integer, m0 + m1 = N 1. We consider a
lemniscate generated by the following polynomial
0
P (z) = P(m0 m1 ) (z) := zm0 (z ; 1)m1 namely
l = l(m0 m1 ) := fz 2 C : jP (z)j = 1g:
Due to l(km0 km1) = l(m0 m1 ) we can suppose that the numbers m0 , m1 are mutually prime.
Note that for N = 1 the lemniscates l(10) , l(01) are the unit circle centered at the point
0 or 1 correspondently. If N 2 then at the unique critical point z0 , z0 6= 0 1, of the
polymomial P (z), i.e. P 0(z0 ) = 0 and P (z0) 6= 0, we have jP (z0)j < 1. Therefore the
lemniscate l consists of the single continuum (see 8]).
Let a = a(m0 m1 ) be a set of all functions a : Z ! f0 1g, such that s(T1a) = l. With
each a 2 a we can associate the following real number from the inteval 0 1]:
qa =
1
X
a(n)
n=0
2n+1 :
We denote by Q(m0 m1 ) the union of all qa when a is scanning over the whole set a. If
N > 1 then the set of all sequences + = fngn0 of permutations of fng S (N~ ) is
uncountable hence the set Q(m0 m1) is uncountable if mj 6= 0, j = 0 1. Consider any
rational number r 2 (0 1) with a nite number of units in its binary reprezentation, i.e.
r=
N
X
rn r = 1:
N0
n+1
n=0 2
0
7
Then choosing b(n) = rn ; a(n), 0 n N0 we derive from the remark 4 that the number
q=r+
1
X
a(n)
n+1
n=N0 +1 2
belongs to Q(m0 m1 ) if a = a~a 2 a(m0 m1 ). This means that the set Q(m0 m1 ) is everywhere
dence in 0 1] for each (m0 m1). On the other hand
Q(m0 m1 )
\
Q(m0 m1 ) = 0
0
if and only if (m0 m1 ) 6= (m00 m01) since l(m0 m1 ) 6= l(m0 m1 ).
Here are few questions which are connected with the problem of description of spectra
of the perturbed shifts
a) Description of properties of the set Q(m0 m1) . In particular is it measurable, or just
zero-measure ?
b) Are Q(10) and Q(01) countable ?
c) What is the complement of the union of all sets Q(m0 m1) on the interval 0 1] ?.
0
0
4 Shift on Sequences of Quaternions and
Cayley Numbers
The proof of our main theorem depends crucially on the product rule for moduli : jzwj =
jzjjwj which holds for all z w 2 C . One can expect that once this condition is fullled,
we may consider the shift on generalized complex numbers: the quaternions H and the
Carley numbers K (see 2], 10]). Note that for these numbers the commutative law
for multiplication does no hold and for the Cayley numbers even the associative law for
multiplication is lost. But still each non-zero element of H or K has an inverse.
We denote by A one of the algebras R C H K . These four algebras are the only nonisomorphic algebras over the real eld of nite dimension with a unit and the product
rule for moduli, i.e.
jabj = jajjbj for all a b 2 A:
Here j j stays for the Euclidean length in R m , m = 1, 2, 4, 8.
Let f be a two sided sequence of elements of the algebra A, f : Z ! A. Consider the
shift operator S and dene the perturbed shift (as above) for a positive number and
given function a : Z ! A
Ta := S + a:
We say that an element of the algebra A belongs to the A-spectrum sA (Ta) of the
operator Ta with a 2 A if and only if the following dierence equation
f (n + 1) + a(n)f (n) = f (n) n 2 Z
8
(10)
has a nontrivial solution and every solution f 2 A is bounded, i.e. jf j := supn jf (n)j < 1.
Remarks. 1. We put 2 A in the equation (10) instead of z 2 C as in the
equation (2) because every nite dimensional complex division algebra with unit element
is isomorphic to C .
2. sC (Ta) = s(Ta ).
3. Let us consider the quaternions H as a four-dimensional real vector subspase of the
matrix space Mat(2 C ) with the matrix multiplication (see 2]), i.e.
H
z ;w :=
: w z 2 C :
w z
Then sH (Ta) coincides with the set of the stability of the dierence system (10) (see
1])
with
f ;f a ;a ; 1
2
1
2
f = f f a = a a = 1 2 2
1
2
1
2
1
fj aj j 2 C j = 1 2:
In this case
jf j2 := det f = jf1j2 + jf2 j2 = sup (jf1(n)j2 + jf2(n)j2):
n2Z
4. Let a(n) a(0) be a stationary sequence. Then the set sA (Ta) is a (m ; 1)dimensional sphere in R m , m = dim A with the radius centered at the point a(0) =
(x1 x2 ::: xm ) 2 R m . In particular sA (S ) = fa 2 A jaj = 1g - the unit sphere.
For given ~a = fa1 a2 ::: aN g 2 AN and a sequence of permutations of the set N~ we
dene a function a~a : Z ! A and a function a~a by the same way as in the case A = C .
The locus of a point for which the product of distances from points of the set ~a 2 AN
is the constant N is a A-lemniscate l~a . Thus, the A-lemniscate l~a is dened by the
equation
l~a = f 2 A : j ; a1j j ; a2 j:::j ; aN j = N g:
Therefore dim l~a = dim A ; 1.
The next statement may be proved word for word as the Main Theorem above (the
case A = C ).
Theorem. For any positive number ,any set ~a 2 AN , and any sequence = fngn2Z
of permutations n of the set N~ = f1 2 ::: N g we have
sA(Ta ) = sA (Ta ) = l~a 0
where a = a~a and a0 = a~a .
Note that all remarks 1 - 5 of the section 3 can be easily formulated in terms of the
algebra A. Any multi-dimensional version of Hilbert Theorem would help description
spectra of general perturbed shift operator.
9
5 Conclusion
The problem of description of spectra of periodic and quasiperiodic dierential operators
belongs to the most challenging problems of the spectral analysis. Our results show that
beyond the frames of selfadjoint and weakly perturbed selfadjoint operators some new
patterns of spectra appear, in particular the spectrum of periodically perturbed shift
operator may ll the boundary of almost any open domain with a compact complement.
On the other hand the problem of description of isospectral deformations of the coecients
admits very natural sollution. The main facts remain true for shifts in all spaces of
sequences of hypercomplex numbers for which the norm of product is equal to the product
of norms (real, complex, Cayley numbers and quaternions).
6 Acknowledgements
.
This research was acomplished at the University of Auckland, New Zealand, and the
rst author (V.O.) is greatful to the Department of Mathematics for excellent working
conditions. The research of the rst author was in part supported by the International
Soros Science Education Program (Grant ISSEP, d98-387). The second author (B.P.)
proudly recognizes the support from Marsden Found of the Royal Society of New Zealand
(Grant 3368152).
References
1] R.P. Agarwal, Dierence Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker, 1992.
2] H.-D. Ebbinghaus et al., Numbers, Springer-Verlag, 1991.
3] D. Hilbert, Uber die Entwicklung einer beliebigen analytischen Funktion einer Variabeln in eine unendliche nach ganzen rationalen Funktionen fortschreitende Reihe,
Gottinger Nachrichten, (1897), pp. 63-70.
4] J.F. Humphreys, A Course in Group Theory, Oxford University Press, 1996.
5] A.I. Markushevich, Theory of Functions of a Complex Variable, v.1, Prentice-Hall,
1965.
6] J.L. Walsh and Helen G. Russell, On the convergence and overconvergence of sequences of polynomials of best simultaneous approximation to several functions analytic in distinct regions, Transaction of the American Mathematical Society, vol.36
(1934), pp. 13-28.
10
7] J.L. Walsh, Lemniscates and equipotential curves of Green's function, American
Mathematical Monthly. vol. 42 (1935), pp. 1-17.
8] J.L. Walsh, The location of critical points of analytic and harmonic functions, American Mathematical Society, 1950.
9] J.L. Walsh, Interpolation and approximation rational functions in the complex domain, 2nd ed., American Mathematical Society, 1956.
10] J.P. Ward, Quaternions and Cayley Numbers, Kluwer Academic Publishers, 1997.
11