Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices
type: Item
instance of: scholarly article
title: Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices
zbMATH Open document ID: 1110.47020
author: Maria Malejki
author: Jan Janas
published in: Journal of Computational and Applied Mathematics
publication date: 2007-01-22T00:00:00Z
review text: The present paper deals with some approaches to study the asymptotic behaviour of eigenvalues of a sufficiently large class of unbounded Jacobi matrices. In particular, the authors prove the following result (Theorem 2.2). Let \(J\) be an Jacobi operator defined on its maximal domain in \(l_{2} (\mathbb{N})\) by \(J = S W + W S^{*} + Q,\) where \(Q = \text{diag} (q_n)\), \(W = \text{diag} (\lambda_{n})\) (\((q_n)\) and \((\lambda_n)\) are given sequences of positive numbers), \(S\) is the canonical shift operator and assume that (i) \(q_n = \rho n^{\alpha} (1 + \Delta_{n})\), \(\Delta_{n} \longrightarrow 0\), \(\rho \in \mathbb{R}\;\backslash \{0\}\); (ii) \(\beta \geq 0\), \(\alpha > 2 \beta + 1\); (iii) \(\Delta_{n + 1} - \Delta_{n} = O(1/n)\). Then the eigenvalues \(\lambda_{n} (J)\) of \(J\) satisfy \(\lambda_{n} (J) = q_{n} + O (1/n^{\alpha-2\beta-1})\) for large \(n\).
reviewed by: Petru A. Cojuhari
Mathematics Subject Classification ID: 47B36
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5117689
zbMATH Keywords: selfadjoint unbounded Jacobi matrix
zbMATH Keywords: point spectrum
zbMATH Keywords: asymptotic behaviour of eigenvalues
MaRDI profile type: Publication
full work available at URL: j.cam.2005.09.033
OpenAlex ID: W2129269734
cites work: Q5445982
cites work: The asymptotical spectrum of Jacobi matrices
cites work: The eigenvalue density of rational Jacobi matrices. II
cites work: Spectral measures and Jacobi matrices related to Laguerre-type systems of orthogonal polynomials
cites work: Absolute continuity for unbounded Jacobi matrices with constant row sums
cites work: Spectra of Jacobi Matrices, Differential Equations on the Circle, and the $su(1,1)$ Lie Algebra
cites work: Q4552807
cites work: Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights.
cites work: Q2752820
cites work: Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries
cites work: Infinite Jacobi Matrices with Unbounded Entries: Asymptotics of Eigenvalues and the Transformation Operator Approach
cites work: Spectral Theory of Jacobi Matrices in $l^2 (\mathbb{Z})$ and the $su(1,1)$ Lie Algebra
cites work: A note on coherent states related to weighted shifts
cites work: Q3873056
cites work: Q3836536
cites work: Q3844537
DOI: 10.1016/J.CAM.2005.09.033
description: scientific article; zbMATH DE number 5117689