On the Discrete Spectrum of Complex Banded Matrices

On the Discrete Spectrum of Complex Banded Matrices

The discrete spectrum of complex banded matrices that are compact perturbations of the standard banded matrix of order p is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The p-banded matr...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2006
Автори: Golinskii, L., Kudryavtsev, M.
Формат: Стаття
Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2006
Назва видання:Журнал математической физики, анализа, геометрии
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/106677
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:On the Discrete Spectrum of Complex Banded Matrices / L. Golinskii, M. Kudryavtsev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 396-423. — Бібліогр.: 10 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-106677
record_format dspace
spelling irk-123456789-1066772016-10-03T03:02:13Z On the Discrete Spectrum of Complex Banded Matrices Golinskii, L. Kudryavtsev, M. The discrete spectrum of complex banded matrices that are compact perturbations of the standard banded matrix of order p is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The p-banded matrix with the discrete spectrum having exactly p limit points on the interval (-2, 2) is constructed. The results are applied to study the discrete spectrum of asymptotically periodic Jacobi matrices. 2006 Article On the Discrete Spectrum of Complex Banded Matrices / L. Golinskii, M. Kudryavtsev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 396-423. — Бібліогр.: 10 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106677 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The discrete spectrum of complex banded matrices that are compact perturbations of the standard banded matrix of order p is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The p-banded matrix with the discrete spectrum having exactly p limit points on the interval (-2, 2) is constructed. The results are applied to study the discrete spectrum of asymptotically periodic Jacobi matrices.
format Article
author Golinskii, L.
Kudryavtsev, M.
spellingShingle Golinskii, L.
Kudryavtsev, M.
On the Discrete Spectrum of Complex Banded Matrices
Журнал математической физики, анализа, геометрии
author_facet Golinskii, L.
Kudryavtsev, M.
author_sort Golinskii, L.
title On the Discrete Spectrum of Complex Banded Matrices
title_short On the Discrete Spectrum of Complex Banded Matrices
title_full On the Discrete Spectrum of Complex Banded Matrices
title_fullStr On the Discrete Spectrum of Complex Banded Matrices
title_full_unstemmed On the Discrete Spectrum of Complex Banded Matrices
title_sort on the discrete spectrum of complex banded matrices
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2006
url http://dspace.nbuv.gov.ua/handle/123456789/106677
citation_txt On the Discrete Spectrum of Complex Banded Matrices / L. Golinskii, M. Kudryavtsev // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 4. — С. 396-423. — Бібліогр.: 10 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT golinskiil onthediscretespectrumofcomplexbandedmatrices
AT kudryavtsevm onthediscretespectrumofcomplexbandedmatrices
first_indexed 2025-07-07T18:51:13Z
last_indexed 2025-07-07T18:51:13Z
_version_ 1837015262129291264
fulltext Journal of Mathematical Physics, Analysis, Geometry 2006, vol. 2, No. 4, pp. 396�423 On the Discrete Spectrum of Complex Banded Matrices L. Golinskii and M. Kudryavtsev Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov, 61103, Ukraine E-mail:golinskii@ilt.kharkov.ua kudryavtsev@ilt.kharkov.ua kudryavstev@onet.com.ua Received November 22, 2005 The discrete spectrum of complex banded matrices that are compact per- turbations of the standard banded matrix of order p is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides �niteness of the discrete spectrum is found. The p-banded matrix with the discrete spectrum having exactly p limit points on the in- terval (�2; 2) is constructed. The results are applied to study the discrete spectrum of asymptotically periodic Jacobi matrices. Key words: banded matrices, discrete spectrum, asymptotically periodic Jacobi matrices. Mathematics Subject Classi�cation 2000: 47B36, 47A10. 1. Introduction In the recent papers [1, 2] I. Egorova and L. Golinskii studied the discrete spec- trum of complex Jacobi matrices such that the operators in `2(N), N := f1; 2; : : :g generated by these matrices are compact perturbations of the discrete laplacian. In turn, these papers are the discrete version of the known Pavlov theorems ([3, 4]) for the di�erential operators of the second order on the semiaxis. The su�cient conditions for the spectrum to be �nite and empty, the domains containing the discrete spectrum and the conditions for the limit sets of the discrete spectrum were found. The goal of this work is to extend the results to the case of operators, generated by banded matrices. The work of the �rst Author was supported in part by INTAS Research Network NeCCA 03-51-6637. c L. Golinskii and M. Kudryavtsev, 2006 On the Discrete Spectrum of Complex Banded Matrices Let us remind that an in�nite matrix D = kdijk 1 i;j=1 is called the banded matrix of order p or just p-banded if dij = 0; ji� jj > p; dij 6= 0; ji� jj = p; dij 2 C : (1.1) According to this de�nition, the Jacobi matrices are banded matrices of order p = 1. Throughout the whole paper we assume that lim i!1 di;i�p = 1; lim i!1 di;i�r = 0; jrj < p; (1.2) and so the operators in `2 = `2(N) generated by matrices (1.1)�(1.2) are compact perturbation of the standard banded operator D0 : di;i�p = 1; dij = 0; ji� jj 6= p; D0 = Sp + (S�)p; (1.3) where S is the one-sided shift operator in `2. It is well known that the spectrum �(D0) of D0 is the closed interval [�2; 2]. According to the Weyl theorem (see, e.g., [5]) the spectrum of the perturbed operator �(D) = [�2; 2] S �d(D), where the discrete spectrum �d(D) is at most denumerable set of points of the complex plane, which are eigenvalues of �nite algebraic multiplicity. All its accumulation points belong to the interval [�2; 2]. Let us denote by ED the limit set for the set �d(D). So, ED = ; means that the discrete spectrum is �nite. Remind that the convergence exponent or Taylor�Besicovitch index of a closed point set F � [�2; 2] is the value �(F ) := inff" > 0 : 1X j=1 jljj " <1g; where fljg are the adjacent intervals of F . De�nition 1.1. We say that the matrix D (1:1) belongs to the class Pp(�), 0 < � < 1, if qn := jdn;n�p � 1j+ p�1X r=�p+1 jdn;n+rj+ jdn;n+p � 1j � C1 exp(�C2 n �) ; (1.4) n 2 N, with the constants C1, C2 > 0, depending on D. The main result of the present paper is the following. Theorem 1.2. Let D 2 Pp(�) where 0 < � < 1 2 . Then ED is a closed point set of the Lebesgue measure zero and its convergence exponent satis�es dimED � �(ED) � 1� 2� 1� � ; (1.5) where dimED is the Hausdor� dimension of ED. Moreover, if D 2 P(1 2 ) then ED = ;, i.e., the discrete spectrum is �nite. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 397 L. Golinskii and M. Kudryavtsev It turns out that the exponent 1=2 in Th. 1.2 is sharp in the following sense. Theorem 1.3. For arbitrary " > 0 and arbitrary points �1; �2; : : : ; �p 2 (�2; 2) there exists an operator D 2 Pp( 1 2 � ") such that its discrete spectrum �d(J) is in�nite and, moreover, ED = f�1; �2; : : : ; �pg: The Theorems 1 and 2 are proved in Sect. 4, where the domains containing �d(D) are also found (under the di�erent assumptions than (1.4)). In Sect. 2 the connection is established between the discrete spectrum and zeros of the determinant constructed of p linearly independent solutions of the linear di�erence equations for the eigenvector. In Sect. 3 the properties of the Jost matrix solutions are studied. Finally, in the last Sects. 5 and 6 the main results are applied to study the spectrum of doubly-in�nite complex banded matrices and the spectrum of the doubly-in�nite asymptotically p-periodic complex Jacobi matrices. 2. The Determinants of Independent Solutions and the Eigenvalues We start out with the equation D~y = �~y (2.1) for generalized eigenvectors ~y = fyngn�1 in the coordinate form:8>>>>< >>>>: d11 y1 + d12 y2 + : : : + d1;p+1 yp+1 = �y1; d21 y1 + d22 y2 + : : : + d2;p+2 yp+2 = �y2; : : : dp;1 y1 + dp;2 y2 + : : :+ dp;2p y2p = �yp; dn;n�pyn�p + dn;n�p+1yn�p+1 + : : : + dn;n+pyn+p = �yn; n = p+ 1; p+ 2; : : : : (2.2) It is advisable to de�ne coe�cients di;j for the indices withmin(i; j) � 0 as follows: dij = 1; ji� jj = p; dij = 0; ji� jj 6= p; (2.3) and so system (2.2) is equivalent to dn;n�p yn�p + dn;n�p+1 yn�p+1 + : : :+ dn;n+p yn+p = �yn; n 2 N; (2.4) with the initial conditions y1�p = y2�p = : : : = y0 = 0: (2.5) 398 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices Thus, the vector ~y = fyngn�1�p 2 `2 is the eigenvector of the operator D corre- sponding to the eigenvalue � if and only if fyng satis�es (2.4), (2.5). It seems natural to analyze equation (2.4) within the framework of the general theory of linear di�erence equations. The equation x(n+ k) + a1(n)x(n+ k � 1) + : : : + ak(n)y(n) = 0 (2.6) is said to belong to the Poincar�e class if ak(n) 6= 0 and there exist limits (in C ) bj = lim n!1 aj(n); j = 1; 2; : : : ; k: Denote by fwjg k j=1 all the roots (counting the multiplicity) of the characteristic equation wk + b1w k�1 + : : :+ bk = 0: (2.7) One of the cornerstones of the theory of linear di�erence equations is the following result due to Perron. Theorem ([6, Satz 3]). Let the roots fwjg of (2:7) lie on the circles �l = fjwj = �lg; l = 1; 2; : : : ;m; �j 6= �k, and exactly vl � 1 of them (counted according to their multiplicity) belongs to each circle �l, so �1 + : : : + �m = k. Then (2:6) has a fundamental system of solutions S = fy1; : : : ; ykg = m[ l=1 Sl ; the sets fSlg are disjoint, jSlj = �l, and for any nontrivial linear combination y(n) of the solutions from Sl lim sup n!1 n p jy(n)j = �l; l = 1; 2; : : : ;m; (2.8) holds. Proposition 2.1. For any � 2 C n[�2; 2] the dimension of the space of `2- solutions of (2:4) equals p. P r o o f. Note that equation (2.4) has order k = 2p (after dividing through by the leading coe�cient dn;n+p) and belongs to the Poincar�e class by assumption (1.2). Its characteristic equation (2.7) has now the form w2p � �wp + 1 = (wp � z)(wp � z�1) = 0 : � = z + z�1; z < 1 : For its roots we have jw1j = : : : = jwpj = jzj < 1 < jzj�1 = jwp+1j = : : : = jw2pj: Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 399 L. Golinskii and M. Kudryavtsev By the Perron theorem, there exists the fundamental system S of the solutions of (2.4) S = fy1; : : : ; yp; yp+1; : : : ; y2pg = S1 [ S2 ; dim spanS1 = dim spanS2 = p ; and each solution y 2 spanS1 is in `2 (and even decreases exponentially fast). Let now y be any solution of (2.4) from `2, y = pX j=1 cjyj + 2pX j=p+1 cjyj = y0 + y00 : But y0 2 `2, and so y00 2 `2 which by (2.8) and jzj�1 > 1 is possible only when cj = 0 for j = p+ 1; : : : ; 2p, as needed. Proposition 2.2. Let fy (i) n gn�1�p, i = 1; 2; : : : ; p, be linearly independent solutions of (2:4) from `2. The number � is an eigenvalue of the operator D if and only if detY0(�) = ������� y (1) 1�p y (1) 2�p : : : y (1) 0 : : : : : : : : : : : : y (p) 1�p y (p) 2�p : : : y (p) 0 ������� = 0: (2.9) P r o o f. Suppose that detY0(�) = 0. Then there are numbers �(1); : : : ; �(p), which do not vanish simultaneously, such that8>< >: �(1)y (1) 1�p + : : : +�(p)y (p) 1�p = 0 ; : : : : : : : : : �(1)y (1) 0 + : : : +�(p)y (p) 0 = 0 : Hence the linear combination yn = �(1)y(1)n + : : :+ �(p)y(p)n ; n � 1� p; (2.10) belongs to `2 and satis�es (2.4), (2.5), i.e., � is an eigenvalue of the operator D. Conversely, let � be an eigenvalue and y = fyngn�1 a corresponding eigenvec- tor. Then fyngn�1�p is an `2-solution of (2.4) with the initial conditions (2.5). By Prop. 2.1 (2.10) holds with coe�cients �(1); : : : ; �(p) which do not vanish si- multaneously. Then (2.9) follows immediately from (2.5). 400 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices 3. The Matrix-Valued Jost Solution The goal of this section is to establish the existence of matrix-valued analogue of the Jost solution for the banded matrix D. Once we have the Jost matrix solution at our disposal, we will be able to construct p linearly independent square- summable solutions of (2.4) and, in view of Prop. 2.2, to reduce the study of the location of the discrete spectrum for the matrix D to the location of the zeros for determinant (2.9), composed of these p solutions. It is convenient to rewrite the initial equation in the form of a three-term recurrence matrix relation, by looking at D as a block-Jacobi matrix. Along this way we can extend the standard techniques of proving the existence of the Jost solution for the Jacobi matrices to the case of banded matrices. De�ne the following p� p-matrices: Ak = 0 B@ d(k�1)p+1; (k�2)p+1 : : : d(k�1)p+1; (k�1)p ... ... dkp; (k�2)p+1 : : : dkp; (k�1)p 1 CA ; Bk = 0 B@ d(k�1)p+1; (k�1)p+1 : : : d(k�1)p+1; kp ... ... dkp; (k�1)p+1 : : : dkp; kp 1 CA ; Ck = 0 B@ d(k�1)p+1; kp+1 : : : d(k�1)p+1; (k+1)p ... ... dkp; kp+1 : : : dkp; (k+1)p 1 CA : (3.1) Then the matrix D can be represented in the form D = 0 BBBBB@ B1 C1 0 0 : : : A2 B2 C2 0 : : : 0 A3 B3 C3 : : : 0 0 A4 B4 : : : : : : : : : : : : . . . . . . 1 CCCCCA ; (3.2) with the upper triangular matrices Ak and the lower triangular Ck which are invertible due to (1.1). To be consistent with (2.3) we put A1 = C0 = I a unit p� p matrix, B0 = 0. Having p solutions f' (l) j gj�1�p, l = 1; 2; : : : ; p of equation (2.4) at hand, we can make up p� p matrices j = 0 BB@ ' (1) (j�1)p+1 : : : ' (p) (j�1)p+1 ... ... ' (1) jp : : : ' (p) jp 1 CCA ; j 2 Z+ := 0; 1; : : : : (3.3) Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 401 L. Golinskii and M. Kudryavtsev and write (2.4) in the matrix form Ak k�1 +Bk k + Ck k+1 = � k; k 2 N: (3.4) It will be convenient to modify equation (3.4), getting rid of the coe�cients Ak's. Suppose that 1X k=1 kI �Akk <1 ; (3.5) where k �k is any norm in the space of matrices. It is well known, that there exists an in�nite product (from the right to the left) A := 1Y j=1 Aj = lim n!1 (AkAk�1 : : : A1) ; and all the matrices A and An are invertible. Denote Lj := 1Y i=j+1 Aj ; LjAj = Lj�1; lim j!1 Lj = I: (3.6) The multiplication of (3.4) from the left by Lk gives Lk�1 k�1 + LkBk k + LkCk k+1 = �Lk k ; Lk�1 k�1 + LkBkL �1 k � Lk k + LkCkL �1 k+1Lk+1 k+1 = �Lk k : Hence the matrices �k := Lk k (3.7) satisfy �k�1 + ~Bk�k + ~Ck�k+1 = ��k ; k 2 N; (3.8) with ~Bk = LkBkL �1 k ; ~Ck := LkCkL �1 k+1 : (3.9) For the de�niteness sake we choose the �row norm� kTk := max 1�k�p pX j=1 jtkjj; T = ftkjg p k;j=1: Then by (3.1) max � kAk � Ik; kBkk; kCk � Ik � � q̂k := max 1�j�p (q(k�1)p+j); k 2 N; (3.10) 402 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices qk are de�ned in (1.4). In accordance with (2.3) q1�p = : : : = q0 = 0, so we put q̂0 = 0. It is clear that 1X k=1 q̂k � 1X k=1 qk � p 1X k=1 q̂k (3.11) so (3.5) holds whenever fqng 2 `1. We will use the complex parameter z related to the spectral parameter � by the Zhukovsky transform: � = z + z�1; jzj < 1 : Denote by g the Green kernel g(n; k; z) = 8< : zk�n � zn�k z � z�1 ; k > n; 0; k � n; n; k 2 Z+ := f0; 1; : : :g; z 6= 0: (3.12) It is clear that g(n; k; z) satis�es the recurrence relations g(n; k + 1; z) + g(n; k � 1; z)� (z + z�1)g(n; k; z) = Æ(n; k); (3.13) g(n� 1; k; z) + g(n+ 1; k; z) � (z + z�1)g(n; k; z) = Æ(n; k); (3.14) where Æ(n; k) is the Kronecker symbol. We proceed with the following conditional result. Proposition 3.1. Suppose that equation (3:8) has a solution Vn with the asymptotic behavior at in�nity lim n!1 Vn(z)z �n = I (3.15) for z 2 D . Then Vn satis�es the discrete integral equation Vn(z) = znI + 1X k=n+1 J(n; k; z)Vk(z); n 2 N; (3.16) with J(n; k; z) = �g(n; k; z) ~Bk + g(n; k � 1; z) � I � ~Ck�1 � : (3.17) P r o o f. Let us multiply (3.13) by Vk, (3.8) for Vk by g(n; k), and subtract the latter from the former g(n; k+1)Vk+g(n; k�1)Vk�g(n; k)Vk�1�g(n; k) ~BkVk�g(n; k) ~CkVk+1 = Æ(n; k)Vk: Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 403 L. Golinskii and M. Kudryavtsev Summing up over k from n to N gives Vn = NX k=n+1 n �g(n; k) ~Bk + g(n; k � 1) � I � ~Ck�1 �o Vk + g(n;N + 1)VN � g(n;N + 1) ~CNVN+1: For jzj < 1 we have by (3.12) and (3.15) lim N!1 � g(n;N + 1)VN � g(n;N) ~CNVN+1 � = znI; which along with J(n; n) = 0 leads to (3.16), as needed. The converse statement is equally simple. Proposition 3.2. Each solution fVn(z)gn�0, z 2 D , of equation (3:16) with n 2 Z+ satis�es the three-term recurrence relation (3:8). P r o o f. Write for n � 1 Vn�1 + Vn+1 = (zn�1 + zn+1)I + 1X k=n J(n� 1; k)Vk + 1X k=n+2 J(n+ 1; k)Vk = (z + z�1)znI + J(n� 1; n)Vn + J(n� 1; n+ 1)Vn+1 + 1X k=n+2 fJ(n� 1; k) + J(n+ 1; k)gVk: By (3.12), (3.17) and (3.14) J(n� 1; n) = � ~Bn; J(n� 1; n+ 1) = �(z + z�1) ~Bn+1 + I � ~Cn and J(n� 1; k) + J(n+ 1; k) = (z + z�1)J(n; k); k � n+ 2: Hence Vn�1 + Vn+1 + ~BnVn � (I � ~Cn)Vn+1 = (z + z�1)znI � ~BnVn � (z + z�1) ~BnVn+1 + (I � ~Cn)Vn+1 + 1X k=n+2 � (z + z�1J(n; k) Vk + ~BnVn � (I � ~Cn)Vn+1 = (z + z�1) zn + 1X k=n+1 J(n; k)vk ! = (z + z�1)Vn; which is exactly (3.8). 404 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices The Jost Solution. To analyze equation (3.16) we introduce new variables ~Vn(z) := z�nVn; ~J(n; k; z) := zk�nJ(n; k; z); (3.18) so that, instead of (3.16), we have ~Vn(z) = I + 1X k=n+1 ~J(n; k; z) ~Vk(z); n 2 Z+: (3.19) Now ~J(n;m; �) is a polynomial with matrix coe�cients. Since jg(n; k; z)zk�nj = jz2(k�n) � 1j jz � z�1j � jzjmin � jk � nj; 2 jz2 � 1j � ; the kernel ~J is bounded by k ~J(n; k; z)k � jzjmin � jk � nj; 2 jz2 � 1j � hk; z 2 D ; (3.20) where hk := k ~Bkk+ kI � ~Ck�1k = kLkBkL �1 k k+ kI �Lk�1Ck�1L �1 k k; k 2 N; (3.21) (see (3.9)). We have hk � kLkkkL �1 k k (kBkk+ kI �Akk+ kAkkkI � Ck�1k) : (3.22) Since kAkk � C(D) and by (3.6) kLkk � kL �1 k k � C(D) (throughout the rest of the paper C = C(D) stands for various positive constants which depend only on p and the original matrix D), we see from (3.10) that hk � C(D)(q̂k�1 + q̂k): (3.23) The existence of the Jost solutions for equation (3.8) will be proved under the assumption X i qi = X i 0 @jdi; i�p � 1j+ jdi; i+p � 1j+ p�1X r=1�p jdi; i+rj 1 A <1 ; which by (3.10), (3.11) and (3.23) impliesX k (kI �Akk+ kBkk+ kI � Ckk) <1 ; X k hk <1: (3.24) The main result concerning equation (3.16) is the following. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 405 L. Golinskii and M. Kudryavtsev Theorem 3.3. (i) Suppose that 1X k=1 qk < 1 ; (3.25) qk are de�ned in (1:4). Then equation (3:16) has a unique solution Vn, which is analytic in D , continuous on D 1 := D n f�1g and * kVn � znIk � Cjzjn ( jzj jz2 � 1j 1X k=n qk ) exp ( Cjzj jz2 � 1j 1X k=n qk ) (3.26) for z 2 D 1 , n 2 Z+. (ii) Suppose that 1X k=1 kqk < 1: (3.27) Then Vn is analytic in D , continuous on D and kVn � znIk � Cjzjn ( 1X k=n kqk ) exp ( C 1X k=n kqk ) ; z 2 D ; n 2 Z+: (3.28) P r o o f. We apply the method of successive approximations. Write (3.19) as Fn(z) = Gn(z) + 1X k=n+1 ~J(n; k; z)Fk(z) (3.29) with Fk(z) := ~Vk(z)� 1; Gn(z) := 1X k=n+1 ~J(n; k; z): (3.30) (i) By (3.20) k ~J(n; k; z)k � �(z)hk ; z 2 D 1 ; �(z) := 2jzjjz2 � 1j�1: (3.31) The series in (3.30) converges uniformly on compact subsets of D 1 by (3.25), (3.24), and so Gn is analytic in D and continuous on D 1 . As a starting point for the method of successive approximation, we put Fn;1 = Gn and denote Fn;j+1(z) := 1X k=n+1 ~J(n; k; z)Fk;j(z): *Following the terminology of selfadjoint case for Jacobi matrices, we call this solution the Jost solution. The function V0 is the matrix analogue of the Jost function. 406 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices Let �0(n) := P 1 k=n+1 hk. By induction on j we prove that kFn;j(z)k � (�(z)�0(n)) j (j � 1)! : (3.32) Indeed, for j = 1 we have Fn;1 = Gn and the result holds by the de�nition of �0 and (3.31). Next, let (3.32) be true. Then jFn;j+1(z)j � �(z) 1X k=n+1 hkkFk;j(z)k � (�(z))j+1 (j � 1)! 1X k=n+1 hk� j 0(k): An elementary inequality (a+ b)j+1 � aj+1 � (j + 1)baj gives 1X k=n+1 hk� j 0(k) � 1 j 1X k=n+1 f�j+10 (k � 1)� � j+1 0 (k)g = � j+1 0 (n) j ; which proves (3.32) for Fn;j+1. Thereby the series Fn(z) = 1X j=1 Fn;j(z) converges uniformly on compact subsets of D 1 and solves (3.29), being analytic in D and continuous on D 1 . It is also clear from (3.32) that kFn(z)k = k ~Vn(z)� Ik � 1X j=1 kFn;j(z)k � �(z)�0(n) expf�(z)�0(n)g: (3.33) To reach (3.26) it remains only to note that by (3.23) �0(n) � C 1X k=n+1 (q̂k�1 + q̂k) � 2C 1X k=n q̂k � 2C 1X j=n qj (the latter inequality easily follows from the de�nition of q̂k). To prove the uniqueness suppose that there are two solutions Fn and ~Fn of (3.29). Take the di�erence and apply (3.31): jFn � ~Fn(z)j = j 1X k=n+1 ~J(n; k; z) � Fk(z)� ~Fk(z) � j; sn � 1X m=n+1 �(z)smhm = rn; (3.34) where sn := kFn(z)� ~Fn(z)k. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 407 L. Golinskii and M. Kudryavtsev Clearly, rn ! 0 as n!1 and if rm = 0 for some m, then by (3.34) we have sn � 0. If rn > 0, then rn�1 � rn rn = sn�(z)hn rn � �(z)hn; rk � MY j=k+1 (1 + �(z)hj) rM (3.35) which leads to rm = 0 and again sn � 0. So the uniqueness is proved. (ii) The same sort of reasoning is applicable with k ~J(n; k; z)k � jzjjk � njhk � khk and kFn;j(z)k � � j 1(n) (j � 1)! ; �1(n) := 1X k=n+1 khk (3.36) instead of (3.31) and (3.32), respectively. We have kFn(z)k = k ~Vn(z)� Ik � 1X j=1 kFn;j(z)k � �1(n) expf�1(n)g (3.37) and �1(n) � C 1X k=n+1 k(q̂k�1 + q̂k) � 2C 1X k=n kq̂k � 2C 1X j=n jqj (the latter inequality easily follows from the de�nition of q̂k). R e m a r k. The constants C that enter (3.23), (3.26) and (3.28) are ine�- cient. This circumstance makes no problem when studying the limit set for the discrete spectrum. In contrast to this case, the e�cient constants are called for when dealing with the domains which contain the whole discrete spectrum. Such constants will be obtained in the next section under additional assumptions of �non asymptotic �avor�. Throughout the rest of the section we assume that condition (3.27) is satis�ed. It is clear that equation (3.8) can be rewritten for the functions ~Vn(z), de�ned in (3.18), as ~Vn(z) = (�� ~Bn)z ~Vn+1(z)� ~Cnz 2 ~Vn+2(z) = (z2 + 1� ~Bnz) ~Vn+1(z)� ~Cnz 2 ~Vn+2(z) : (3.38) Let us now expand ~Vn(z) in the Taylor series taking into account de�nition (3.18) and (3.26) ~Vn(z) = I + 1X j=1 K(n; j)zj : (3.39) 408 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices Here kK(n; j)k1n;j=1 is the operator which transforms the Jost solutions of (2.1) for D = D0 to that of (2.1) for D. If we plug (3.39) into (3.38) and match the coe�cients for the same powers zj we have j = 1 : K(n; 1) = K(n+ 1; 1)� ~Bn ; j = 2 : K(n; 2) = I +K(n+ 1; 2) � ~BnK(n+ 1; 1) � ~Cn ; j � 2 : K(n; j + 1) = K(n+ 1; j � 1) +K(n+ 1; j + 1)� ~BnK(n+ 1; j) � ~CnK(n+ 2; j � 1): Summing up each of these expressions for k = n; n+1; : : : , it is not hard to verify that K(n; 1) = � 1X k=n+1 ~Bk�1 ; (3.40) K(n; 2) = � 1X k=n+1 n ~Bk�1K(k; 1) + ( ~Ck�1 � I) o ; (3.41) K(n; j+1) = K(n+1; j�1)� 1X k=n+1 n ~Bk�1K(k; j) + � ~Ck�1 � I � K(k + 1; j � 1) o : (3.42) In the last step we used K(n; j) ! 0 for n ! 1 and any �xed j, which follows from the Cauchy inequality and (3.28) kK(n; j)k � max k ~Vn(z)� Ik � C 1X k=n kqk : From (3.40)�(3.42), using the induction on j, we obtain kK(n; j)k � �(n; j)� � n+ � j 2 �� ; n 2 Z+; (3.43) where �(n) and �(n;m) are de�ned by �(n) := 1X j=n gj ; �(n;m) := n+m�1Y j=n+1 (1 + �(j)) = m�1Y j=1 (1 + �(n+ j)) ; (3.44) and gj = k ~Bjk+ kI � ~Cjk: (3.45) Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 409 L. Golinskii and M. Kudryavtsev In fact, for j = 1 we have �(n; 1) = 1, � 1 2 � = 0 and kK(n; 1)k � �(n + 1) � �(n). Further, for j = 2, kK(n; 2)k � 1X k=n+1 n k ~Bk�1kkK(k; 1)k + k ~Ck�1 � Ik o � 1X k=n+1 n (kK(k; 1)k + 1) � k ~Bk�1k+ k ~Ck�1 � Ik � � 1X k=n+1 gk (1 + �(k)) � (1 + �(n+ 1)) �(n+ 1) = �(n; 2)�(n + 1): When we pass from the even j = 2l to the odd 2l + 1, we have by (3.42) and by the inductive hypothesis: kK(n; 2l + 1)k � �(n+ 1; 2l � 1))�(n + l) + 1X k=n+1 n k ~Bk�1kkK(k; 2l)k + k ~Ck�1 � IkkK(k + 1; 2l � 1)k o : But, according to the inductive hypothesis, K(k; 2l)k � �(k; 2l)�(k + l) ; kK(k + 1; 2l � 1)k � �(k + 1; 2l � 1)�(k + l) � �(k; 2l)�(k + l): Using these inequalities, we obtain kK(n; 2l + 1)k � �(n+ 1; 2l � 1)�(n+ l) + 1X k=n+1 dk�(k; 2l)�(k + l) � �(n+ 1; 2l � 1)�(n+ l) + �(n+ l)�(n+ 1; 2l)�(n + 1) = �(n+ l) f�(n+ 1; 2l � 1) + �(n+ 1; 2l)�(n + 1)g : But �(n+ 1; 2l � 1) + �(n+ 1; 2l)�(n + 1) = j=n+2l�1Y j=n+2 (1 + �(j)) + j=n+2lY j=n+2 (1 + �(j)) �(n+ 1) = j=n+2l�1Y j=n+2 (1 + �(j)) f1 + (1 + �(n+ 2l)) �(n+ 1)g � j=n+2l�1Y j=n+2 (1 + �(j)) f1 + �(n+ 2l) + (1 + �(n+ 2l)) �(n+ 1)g = j=n+2l�1Y j=n+2 (1 + �(j)) f(1 + �(n+ 2l)) (1 + �(n+ 2l))g = �(n; 2l + 1); 410 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices from which we have kK(n; 2l + 1)k � �(n+ 2l)�(n+ l); as needed. Analogous calculations help us to pass from the odd j = 2l+ 1 to the even j + 1 = 2l + 2. It is easy to see that �(n) and �(n;m) in (3.44) can be replaced by ~�(n) := C 1X j=n qj; ~�(n;m) := n+m�1Y j=n+1 (1 + ~�(j)) = m�1Y j=1 (1 + ~�(n+ j)) : (3.46) Further, it is evident that f�(n)g 2 `1 and the sequences ~�(�) and ~�(�;m) decrease monotonically. Hence kK(n;m)k � 1Y j=1 (1 + ~�(j)) 1X k=n+[m 2 ] qk: (3.47) Taking the latter expression with n = 0, we come to the following. Theorem 3.4. Under hypothesis (3:27) the Taylor coe�cients of the matrix- valued function �(z) := V0(z) = 1X n=0 Æ(n)zn (3.48) admit the bound kÆ(n)k � C 1Y j=1 (1 + ~�(j)) 1X k=[n 2 ] qk; (3.49) where [x] is an integer part of x. In particular, � belongs to the space W+ of absolutely convergent Taylor matrix-valued series. Let us now denote by �ij the entries of the matrix �: � := k�ijk p i;j=1 : Corollary 3.5. Let for the banded matrix D the numbers Mn+1 := 1X k=0 (k + 1)n+1qk <1: (3.50) Then the n-th derivative �(n)(z) = V (n) 0 (z) belongs to W+ and max z2D ����(n) ij (z) ��� � C(D) 4n n+ 1 Mn+1; i; j = 1; 2 : : : ; p: (3.51) Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 411 L. Golinskii and M. Kudryavtsev P r o o f. The statement is a simple consequence of (3.50), the series expansion �(n)(z) = 1X j=0 (j + 1) : : : (j + n)Æ(j + n)zj ; bounds (3.49) and the obvious inequality j� (n) ij (z)j � k�(n)k. 4. The Limit Set and Location of the Discrete Spectrum Consider the matrix-valued Jost solutions Vk(z), which exist 8z 2 C nf0g. The p scalar solutions fv (l) j gj�1�p, l = 1; 2; : : : ; p of equation (2.4), constructed from Vk(z) by formulae (3.7 and (3.3), are linearly independent and belong to `2 due to asymptotic formula (3.26). According to Prop. 2.2, the number � = �+��1 is an eigenvalue for the operator D if and only if the determinant of the matrix- valued function �(z) = V0(z) vanishes at the point �. Thus, the study of the discrete spectrum of the operator D is reduced to the study of zeros of the function (z) := det�(z): (4.1) The main topic considered in this section is the limit set ED of the discrete spectrum of the operator D. Remind that ED � [�2; 2]. Let D 2 Pp(�) (see Def. 1.1). Since kLkk and kL�1k k are uniformly bounded, it is clear from 3:24 that hn, de�ned in (3.21), satis�es the same inequality: hn � C1 exp(�C2 (n+ 1)�) (4.2) (with the same exponent �, but other constants C1; C2 > 0). We are in a position now to prove the �rst result announced in the introduc- tion. Theorem 4.1. Let D 2 P(�) where 0 < � < 1 2 . Then ED is a closed point set of the Lebesgue measure zero and its convergence exponent satis�es dimED � �(ED) � 1� 2� 1� � ; (4.3) where dimED is the Hausdor� dimension of ED. Moreover, if J 2 P(1 2 ) then ED = ;, i.e., the discrete spectrum is �nite. P r o o f. Denote by A the set of all functions, analytic inside D and continuous in D . Recall that the set E on the unit circle T is called a zero set for a class X � A of functions, if there exists a nontrivial function f 2 X , which vanishes on E. We want to show that the function belongs to a certain class 412 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices X (the Gevr�e class, see below) with the known properties of its zero sets. Note that, since X � A, then, according to the Fatou theorem, the zero set has the Lebesgue measure zero. We begin with certain bounds for the derivatives of the function , which can be obtained from Th. 3.4 and Cor. 3.5. For this we write (z) := det�(z) = X � sign��1;�(1)�2;�(2) : : :�p;�(p) ; where � are the permutations of the set f1; 2; : : : ; pg. For the n-th derivative of (n)(z) = X � sign� X Pp 1 kj=n kj�0 � n k1; k2; : : : ; kp � � (k1) 1;�(1) � (k2) 2;�(2) : : :� (kp) p;�(p) (4.4) holds, where the multinomial coe�cients � n k1; k2; : : : ; kp � are de�ned from the identity (x1 + x2 + : : :+ xp) p = X Pp 1 kj=n kj�0 � n k1; k2; : : : ; kp � xk11 xk22 : : : x kp p : (4.5) (4.5) with x1 = x2 = : : : = xp = 1 gives X Pp 1 kj=n kj�0 � n k1; k2; : : : ; kp � = pn : From (4.4) and (3.51) we immediately derive max z2D ��� (n)(z)��� � Cpp! 4n X Pp 1 kj=n kj�0 � n k1; k2; : : : ; kp � Mk1+1Mk2+1 : : : Mkp+1; (4.6) where the numbers Mr are de�ned in (3.50). To estimate the product Mk1+1 : : :Mkp+1 note �rst, that by (4.2) Mr � C 1X k=0 (k + 1)r exp � � C 2 (k + 1)� � exp � � C 2 (k + 1)� � : An elementary analysis of the function u(x) = xr exp(�C 2 x�) gives max x�0 u(x) = u(x0) = � 2r C� �r=� e�r=� = � 2 C�e �r=� rr=�; x0 = � 2r C� �1=� ; Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 413 L. Golinskii and M. Kudryavtsev so that Mr � B � 2 C�e �r=� rr=�; B = C 1X k=0 exp � � C 2 (k + 1)� � : Hence, Mk1+1 : : : Mkp+1 � Bp � 2 C�e �n+p � (k1 + 1) k1+1 � : : : (kp + 1) kp+1 � : The inequalities � n+ 1 n �n=� < e1=� ; (n+ 1)1=� < e(n+1)=� ; (n+ 1)(n+1)=� < e1=�nn=�e(n+1)=� lead to the bound Mk1+1 : : :Mkp+1 � B2 � 2 C� �n=� k k1=� 1 : : : k kp=� p : Substituting the latter into (4.6) gives max z2D j (n)(z)j � C4npn � 2 C� �n=� nk1=� : : : nkp=� � C ~Cnnn=�; n � 0: (4.7) In other words, the function for D 2 P(�) belongs to the Gevr�e class G�. The rest of the proof goes along the same line of reasoning as in [2]. Suppose �rst that 0 < � < 1=2. The celebrated Carleson's theorem [7] gives a complete description of zero sets for G�. It claims that 1X j=1 jlj j 1�2� 1�� <1; where fljg are adjacent arcs of the zeros set of . Hence the right inequality in (4.3) is obtained. The left inequality is a general fact of the fractal dimension theory. When � = 1=2 the Gevr�e class G1=2 is known to be quasi-analytic, i.e., it doesn't contain nontrivial function f , such that f (n)(�0) = 0 for all n � 0 and some �0 2 T. So each function f 2 G1=2 may have only �nite number of zeros inside the unit disk, which, due to the relation between the discrete spectrum of D and zeros of , proves the second statement of the theorem. It turns out that the exponent 1=2 in Th. 1 is sharp in the following sense. 414 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices Theorem 4.2. For arbitrary " > 0 and di�erent points �2 < �i < 2, i = 1; : : : ; p, there exists an operator D 2 Pp( 1 2 � ") such that its discrete spectrum �d(D) is in�nite and, moreover, ED = f�1; : : : ; �pg. P r o o f. The proof of the theorem is based on the similar result, proved in [2] for the Jacobi matrices, that is, for p = 1. If p > 1 than given �1; : : : ; �p we construct the complex Jacobi matrices J(i) = 0 BBB@ b0(i) a0(i) a0(i) b1(i) a1(i) a1(i) b2(i) a2(i) . . . . . . . . . 1 CCCA ; aj(i) > 0; bj(i) 2 R; j � 1; i = 1; 2; : : : ; p; (4.8) which belong to the class P1( 1 2 � ") and have the only point of accumulation of the discrete spectrum EJ(i) = �i. Consider now the p-banded matrix D = kdijk 1 i;j=1 with dpn+i; pn+i = bn(i); dpn+i; p(n+1)+i = dp(n+1)+i; pn+i = an(i); dpn+i; pn+i+j = dpn+i+j; pn+i = 0; j 6= 0; p; i = 1; : : : ; p: Since J(i) 2 P1( 1 2 � "), it is easily seen that D 2 Pp( 1 2 � "). The space `2, where the operator D acts, can be decomposed as follows: `2 = pM i=1 Li with Li = Linfêi; êp+i; ê2p+i; : : : ; ênp+i; : : :g and fêkgk2N the standard basis vec- tors in `2. It is shown directly that the subspaces Li are invariant for D. The restriction Di = D jLi ofD on the subspace Lj has the matrix representa- tion J(i). Since D = Lp i=1Di and every operator Di has a discrete spectrum with the only accumulation point �i, the discrete spectrum D is the union of the dis- crete spectra of the operators J(i) and it has p accumulation points �1; �2; : : : ; �p, as needed. The second issue we address in this section concerns the domains which contain the whole discrete spectrum, and conditions for the lack of the discrete spectrum. We begin with (3.33) and (3.37) for n = 0 k�(z)�Ik � 2�0(0) jz � z�1j exp � 2�0(0) jz � z�1j � ; k�(z)�Ik � �1(0) expf�1(0)g ; (4.9) with � = ~V0, �0(0) = P 1 k=1 hk, �1(0) = P 1 k=1 khk, hk de�ned in (3.21), which hold under assumptions (3.25) and (3.27), respectively. To work with (4.9) we Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 415 L. Golinskii and M. Kudryavtsev have to �nd the e�cient constant C which enters (3.23). To do this let us go back to (3.10) and assume that q := sup n�1 qn = sup n�1 � jdn;n�p � 1j+ jdn;n+p � 1j+ p�1X r=�p+1 jdn;n+rj � < 1; (4.10) which now implies sup k�1 kAk � Ik � q; sup k�1 kAkk � 1 + q < 2: (4.11) It is not hard to show now that supk�1 kA �1 k k � (1� q)�1, sup k�1 kLkk � expf 1X j=1 kAj � Ikg; sup k�1 kL�1k k � exp n 1 1� q 1X j=1 kAj � Ik o and sup k�1 kLkkkL �1 k k � exp n2� q 1� q 1X j=1 kAj � Ik o � exp n2� q 1� q Q0 o ; Q0 := 1X k=1 qk : Hence, instead of (3.23), we have by (3.22) hk � 2 exp n2� q 1� q Q0 o (q̂k�1 + q̂k) and so �0(0) � 4 exp n2� q 1� q Q0 o Q0; �1(0) � 4 exp n2� q 1� q Q0 o Q1; Q1 := 1X k=1 kqk: (4.12) Let now t be a unique root of the equation tet = 1; t � 0:567: (4.13) Theorem 4.3. Assume that Q0 <1 and q < 1. Then the domain G(D) = fz + z�1 : z 2 g with := fz 2 D : jz � z�1j > 8Q0 t exp n2� q 1� q Q0 o is free from the discrete spectrum �d(D). Moreover, assume that Q1 <1. Then D has no discrete spectrum as long as exp n2� q 1� q Q0 o Q1 < 4 t : 416 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices P r o o f. It is clear from the �rst inequality in (4.9) that k�(z) � Ik < 1 (and the more so, det�(z) 6= 0) whenever jz�z�1j > 2�0(0)t �1. The rest follows immediately from the �rst bound in (4.12) and the relation between eigenvalues of D and zeros of �. The second statement is proved in exactly the same way by using the second inequality in (4.9) and the second bound in (4.12). R e m a r k. Suppose that c = 8Q0 t exp n2� q 1� q Q0 o < 2: Then �d(D) is contained in the union of two symmetric rectangles �d(D) � � � : p 4� c2 < jRe �j < p 4 + c2; jIm�j < c2 4 � : 5. The Discrete Spectrum of Doubly-In�nite Banded Matrices A doubly-in�nite complex matrix D = kdijk 1 i;j=�1 is called the banded matrix of order p if dij = 0; ji� jj > p; dij 6= 0; ji� jj = p; dij 2 C : (5.1) There is a nice �doubling method� that relates a doubly-in�nite p-banded matrix to a certain semi-in�nite banded matrix of order 2p. Indeed, let fekgk2Z be the standard basis in `2(Z). Consider a transformation U : `2(Z) �! `2(N), de�ned by Uek = ê�2k; k < 0; Uek = ê2k+1; k � 0; where fêkgk2N is the standard basis in `2(N). The transformation U is clearly isometric and it is easy to check directly that the matrix D̂ := UDU�1 (5.2) is the semi-in�nite band matrix of order 2p unitarily equivalent toD. For instance, in the case of Jacobi matrices (p = 1) J = 0 BBBBBB@ . . . . . . . . . a�1 b�1 c0 a0 b0 c1 a1 b1 c2 . . . . . . . . . 1 CCCCCCA ; Ĵ = 0 BBB@ B̂0 Ĉ1 Â1 B̂1 Ĉ2 Â2 B̂2 Ĉ3 . . . . . . . . . 1 CCCA Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 417 L. Golinskii and M. Kudryavtsev with B̂0 = � b0 a0 c0 b�1 � ; B̂k = � bk 0 0 b�k�1 � ; Âk = � ak 0 0 c�k � ; Ĉk = � ck 0 0 a�k � ; k 2 N: Similarly to the semi-in�nite case (1.4), we put qn := jdn;n�p � 1j+ p�1X j=�p+1 jdn;n+jj+ jdn;n+p � 1j; m 2 Z: (5.3) We say that the matrix D belongs to the class Pp(�), 0 < � < 1, if qn � C1 exp(�C2 jnj �); C1; C2 > 0; n 2 Z: (5.4) It is clear from the construction that the semi-in�nite matrix D̂ (5.2) belongs to P2p(�) whenever the doubly-in�nite matrix D belongs to Pp(�). So the results of the previous section can be easily derived for doubly-in�nite banded matrices as well. For instance, the following statement holds. Theorem 5.1. Let D 2 Pp(�) where 0 < � < 1 2 . Then ED is a closed point set of the Lebesgue measure zero and its convergence exponent satis�es dimED � �(ED) � 1� 2� 1� � ; (5.5) where dimED is the Hausdor� dimension of ED. Moreover, if D 2 Pp( 1 2 ) then ED = ;, i.e., the discrete spectrum is �nite. 6. The Discrete Spectrum of Asymptotically Periodic Jacobi Matrices The goal of this section is to apply the results obtained above to the study of the spectrum of doubly-in�nite asymptotically periodic complex Jacobi matrices. De�nition 6.1. A doubly-in�nite complex Jacobi matrix J0 = 0 BBBBBB@ . . . . . . . . . a0 �1 b0 �1 c00 a00 b00 c01 a01 b01 c02 . . . . . . . . . 1 CCCCCCA ; b00 = (J0e0; e0); (6.1) a0nc 0 n 6= 0, a0n; b 0 n; c 0 n 2 C , is called p-periodic if a0n+p = a0n; b0n+p = b0n; c0n+p = c0n; n 2 Z: 418 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices The following result due to P.B. Naiman [8, 9] is the key ingredient in our argument. Theorem. Each p-periodic Jacobi matrix J0 satis�es the algebraic equation of the p-th degree P (J0) = Ep; where Ep = fejkg is a p-banded matrix with the entries eij = 8>>>>>>< >>>>>>: � := pY k=1 a0k; i� j = p; Æ := pY k=1 c0k; i� j = �p; 0; ji� jj 6= p: (6.2) Next, let � := P (�1)(�) = fz 2 C : P (z) 2 �g be the preimage of the ellipse in the complex plane � := fz 2 C : z = �eipt + Æe�ipt; 0 � t � 2� p g: Then the spectrum �(J0) = �. The polynomial P is known as the Burchnall�Chaundy polynomial for J0 and given explicitly in [9, p. 141]: P (�) = det � �� J(1; p) � � a1c1 det � �� J(2; p� 1) � ; J(m;n) := 0 BBBB@ b0m c0m+1 a0m+1 b0m+1 . . . . . . . . . c0n a0n b0n 1 CCCCA : There is also another way to exhibit the spectrum of any p-periodic Jacobi matrix J0. Consider the p� p-matrix �p(t) := 0 BBBB@ b01 c02 a01e ipt a02 b02 . . . . . . . . . c0p c01e �ipt a0p b0p 1 CCCCA : Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 419 L. Golinskii and M. Kudryavtsev with the eigenvalues f�j(t)g p j=1, and put range�j := fw 2 C : w = �j(t); 0 � t � 2� p g: Then �(J0) = p[ j=1 range�j : Indeed, it is easy to compute det (�(t)� �) = (�1)p � P (�)� �eipt � Æe�ipt � : For further information about the spectra of the periodic Jacobi matrices with algebro-geometric potential see [10]. For the rest of the paper we restrict our consideration by the so called quasi- symmetric matrices with pY k=1 a0k = pY k=1 c0k; � = Æ: (6.3) The latter certainly holds for the symmetric periodic matrices (a0n = c0n). In this case we have Q(J0) = D0 with D0 de�ned in (1.3) for the polynomial Q = ��1P , and �(J0) = Q(�1) � [�2; 2] � : (6.4) It is not hard to make sure that �(J0) is a collection of �nitely many algebraic arcs with no closed loops, that is, the complement C n�(J0) is a connected set. Indeed, if any of these arcs formed a closed loop �1 � � with the interior domain G1, then =Q would vanish on �1 and so =Q � 0 in G1, which is impossible, unless Q is a real constant. Let now J be a complex asymptotically p-periodic doubly-in�nite Jacobi ma- trix with the quasi-symmetric p-periodic background J0 (6.1) J = 0 BBBBBB@ . . . . . . . . . a�1 b�1 c0 a0 b0 c1 a1 b1 c2 . . . . . . . . . 1 CCCCCCA ; ancn 6= 0; (6.5) and !n := jan � a0nj+ jbn � b0nj+ jcn+1 � c0n+1j ! 0; k ! �1: 420 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices In other words, J�J0 is a compact operator in `2(Z). Since C n�(J0) is connected, a version of the Weyl theorem holds (Cf. [5], Lem. I.5.2) �(J) = �(J0) [ �d(J); where the discrete spectrum �d(J) is an at most denumerable set of eigenvalues of �nite algebraic multiplicity o� �(J0), which can accumulate only to �(J0). We proceed with two simple propositions. Proposition 6.2. Let Dm1 and Dm2 be two banded matrices of orders m1 and m2, respectively. Then 1. D = Dm1 Dm2 is the banded matrix of order m = m1 +m2. 2. If m1 < m2 then Dm1 +Dm2 is the banded matrix of order m2. In particular, if T is a polynomial of degree p and D is a banded matrix of order m, then T (D) is the banded matrix of order pm. P r o o f. (2) is obvious. To prove (1) let us show that the elements of the extreme diagonals of D do not vanish. Indeed, let Dml = fd (l) ij g, l = 1; 2, and D = fdijg. Then dk; k+m = 1X j=1 d (1) k j d (2) j; k+m = d (1) k k+m1 d (2) k+m1; k+m 6= 0; k 2 N: Similarly, dk;k�m = d (1) k;k�m1 d (2) k�m1;k�m 6= 0. The rest is plain. Proposition 6.3. Let J1 and J2 be two bounded (with bounded entries) com- plex Jacobi matrices. Put !k = X j j(J1 � J2)kj j = X j jh(J1 � J2)ej ; ekij : Then for any polynomial T of degree p we have X j j(T (J1)� T (J2))kj j = X j jh(T (J1)� T (J2))ej ; ekij � C k+pX s=k�p !s; where a positive constant C depends on J1, J2 and T . Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 421 L. Golinskii and M. Kudryavtsev P r o o f. For T (z) = zn the statement follows immediately from the banded structure of the powers Jml , (J�l ) s, l = 1; 2 and the equality Jn1 � Jn2 = n�1X j=0 J n�j�1 1 (J1 � J2)J j 2 : The rest is straightforward. De�nition 6.4. We say that the matrix J (6:5) belongs to the class P(�; J0), if !n � C1 exp(�C2 jnj �); 0 < � < 1; C1; C2 > 0; n 2 Z: (6.6) Our main result claims that �d(J) is a �nite set as long as J 2 P(1=2; J0). Theorem 6.5. Let J be an asymptotically p-periodic doubly-in�nite Jacobi ma- trix (6:5) with the quasisymmetric background J0 (6:1), (6:3). If J 2 P(1=2; J0), then �(J) is the union of p algebraic arcs and a �nite number of eigenvalues of the �nite algebraic multiplicities o� these arcs. P r o o f. Let T = Q = ��1P be the Burchnall�Chaundy polynomial for J0. By Prop. 6.2 Q(J) is the p-banded matrix. Since J 2 P(1=2; J0) the matrix Q(J) is close to Q(J0) = D0 with D0 de�ned in (1.3) in the sense of Prop. 6.3X j j(Q(J)�D0)kjj � C1e �C2jkj 1=2 and so Q(J) 2 Pp(1=2). According to Th. 5.1 �(Q(J)) = [�2; 2] [ E, where the set E of the eigenvalues o� [�2; 2] is now �nite. On the other hand, as we know, �(J) = � [ F , where the set F of the eigenvalues o� � = �(J0) is at most denumerable, and � = Q(�1)([�2; 2]). Therefore, by the Spectral Mapping Theorem Q(F ) = E and so F is a �nite set, as claimed. Acknowledgements. The Authors thank Prof. M. Pituk for his helpful com- ments on the asymptotic behavior of the solutions of linear di�erence equations and for the reference to [6]. References [1] I. Egorova and L. Golinskii, On Location of Discrete Spectrum for Complex Jacobi Matrices. � Proc. AMS 133 (2005), 12, 3635�3641. [2] I. Egorova and L. Golinskii, On limit sets for the Discrete Spectrum of Complex Jacobi Matrices. � Mat. Sb. 196 (2005), 6, 43�70. (Russian) 422 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 On the Discrete Spectrum of Complex Banded Matrices [3] B.S. Pavlov, On Nonselfadjoint Schr�odinger Operator I. � Probl. Math. Phys., LGU 1 (1966), 102�132. (Russian) [4] B.S. Pavlov, On Nonselfadjoint Schr�odinger Operator II. � Probl. Math. Phys., LGU 1 (1966), 102�132. (Russian) [5] I. Gohberg and M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. Nauka, Moscow, 1965. (Russian) [6] O. Perron, �Uber Summengleichungen und Poincar�esche Di�erenzengleichingen. � Math. Annal. 84 (1921), 1�15. (German) [7] L. Carleson, Sets of Uniqueness for Functions Analytic in the Unit Disc. � Acta Math. 87 (1952), 325�345. [8] P.B. Naiman, On the Theory of Periodic and Limit-Periodic Jacobi Operators. � Sov. Math. Dokl. 3 (1962), 4, 383�385. (Russian) [9] P.B. Naiman, To the Spectral Theory of the Non-Symmetric Periodic Jacobi Ma- trices. � Notes Depart. Math. and Mech. Kharkov State Univ. and Kharkov Math. Soc. XXX (1964), 4, 138�151. (Russian) [10] V. Batchenko and F. Gesztesy, On the Spectrum of Jacobi Operators with Quasi-periodic Algebro-Geometric Coe�cients. � Int. Math. Res. Papers 10 (2005), 511�563. Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 4 423

Схожі ресурси