Difference equations of second order with spectral singularities
type: Item
instance of: scholarly article
title: Difference equations of second order with spectral singularities
zbMATH Open document ID: 1025.39001
DOI: 10.1016/S0022-247X(02)00655-8
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-04-28T00:00:00Z
review text: Spectral properties of the discrete eigenvalue problem \[ a_{n}y_{n+1}+b_ny_n+a_{n-1}y_{n-1}=\lambda y_n,\quad n\in \mathbb Z, \tag{*} \] where \(a,b\) are complex sequences with \(a_n\neq 0\) satisfying \(\sum_{n\in \mathbb Z}|n|(|1-a_n|+|b_n|)<\infty \), and \(\lambda\) is the spectral parameter, are investigated. Eigenvalues and spectral singularities of (*) are defined via zeros of the Wronskian of solutions of this equation. The main result of the paper reads as follows. Theorem: Suppose that for some \(\varepsilon>0\) and \(1/2\leq \delta<1\) \[ \sup_{n\in \mathbb Z}\left\{e^{\varepsilon|n|^{\delta}}\left(|1-a_n|+|b_n|\right)\right\}<\infty. \] Then (*) has only a finite number of eigenvalues and spectral singularities and each of them is of finite multiplicity. A related paper of the same authors is published in [J. Math. Anal. Appl. 261, 461-478 (2001; Zbl 0992.39018)], where the statement of Theorem is proved under a stronger assumption.
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1901866
zbMATH Keywords: difference equations
zbMATH Keywords: spectral analysis
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: spectral singularities
author: Murat Adivar
author: Elgiz Bairamov
reviewed by: Ondřej Došlý
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00655-8
OpenAlex ID: W1982937183
description: scientific article; zbMATH DE number 1901866