Classification of non-self-adjoint singular Sturm-Liouville difference equations
type: Item
instance of: scholarly article
title: Classification of non-self-adjoint singular Sturm-Liouville difference equations
zbMATH Open document ID: 1228.39006
author: Huaqing Sun
author: Haibin Jing
author: Jiangang Qi
published in: Applied Mathematics and Computation
publication date: 2011-06-28T00:00:00Z
review text: This paper is concerned with the Sturm-Liouville difference equation with complex coefficients: \[ -\nabla \left( p(n)\Delta x(n)\right) +q(n)x(n)=\lambda w(n)x(n),\;n\in {\mathbb{N}}_{0}=\{0,1,2,\dots\}, \tag{1} \] where \(\left\{ p(n)\right\} _{n=-1}^\infty, \left\{q(n)\right\}_{n=0}^{\infty}\) are complex sequences such that \(p(n)\neq 0\) for \( n=0,1,2,\dots\), \(\left\{ w(n)\right\} _{n=0}^{\infty }\) is a positive real sequence, and \(\lambda \) is a spectral parameter. An important question is whether a solution \(\left\{ x(n)\right\} _{n=-1}^{\infty }\) (which depends on \(\lambda \)) is square summable, that is, whether the series \( \sum_{n=0}^{\infty }w(n)\left| x(n)\right| ^{2}\) is finite. Let \[ \Omega =\overline{co}\left\{ \frac{q(n)}{w(n)}+rp(n),\;n\in {\mathbb{N}}_{0},\;0<r<\infty \right\}, \] where \(\overline{co}\) stands for the closed convex hull, let \[ S=\left\{ (\theta ,K):K\in \partial \Omega ,\text{Re}\left| (z-K)e^{i\theta }\right| \geq 0\text{ for all }z\in \Omega \right\}, \] and let \[ \Lambda _{\theta ,K}=\left\{ \lambda \in {\mathbb{C}}:\text{Re}\left( \left( \lambda -K\right) e^{i\theta }\right) <0\right\}. \] The authors give the following definition: Let \((\theta ,K)\in S\) and \(\lambda \in \Lambda _{\theta,K}\). If (1) has exactly one linearly independent solution \(\left\{x(n)\right\}\) satisfying \[ \sum_{n=0}^{\infty }\text{Re}\left[ e^{i\theta }\left( q(n)-Kw(n)\right) \right] \left| x(n)\right| ^{2}+\sum_{n=0}^{\infty }\text{Re}\left( e^{i\theta }p(n)\right) \left| \Delta (n)\right| ^{2}+\sum_{n=0}^{\infty }w(n)\left| x(n)\right| ^{2}<\infty, \tag{2} \] and this is the only linearly independent square summable solution, then (1) is of Type I. If (1) has exactly one linearly independent solution satisfying (2), but all solutions of (1) are square summable, then (1) is of Type II. If all solutions of (1) satisfy (2) (and hence square summable), then (1) is of Type III. It is shown that a Type I equation does not depend on \(…
reviewed by: Sui Sun Cheng
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 5914115
zbMATH Keywords: Sturm-Liouville difference equation
zbMATH Keywords: Hamiltonian difference system
zbMATH Keywords: square summable solition
zbMATH Keywords: limit point
zbMATH Keywords: limit circle
zbMATH Keywords: classification scheme
MaRDI profile type: Publication
full work available at URL: j.amc.2011.02.107
OpenAlex ID: W2091067078
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DOI: 10.1016/J.AMC.2011.02.107
description: scientific article; zbMATH DE number 5914115