On the spectra of infinite-dimensional Jacobi matrices
type: Item
instance of: scholarly article
title: On the spectra of infinite-dimensional Jacobi matrices
zbMATH Open document ID: 0671.47021
DOI: 10.1016/0021-9045(88)90022-6
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(\ell_ 2\supset D(J)\to^{J}\ell_ 2\) be a self-adjoint operator with the representation \[ Je_ n=a(n+1)e_{n+1}+b(n)e_ n+a(n)e_{n- 1},\quad (n=1,2,\dots);\quad Je_ 0=a(1)e_ 1+b(0)e_ 0. \] By using the Green function, a general comparison equation is constructed; and this equation is invoked to obtain an upper bound on the number of eigenvalues of \(J\) using Bargmann's argument. Also, by using a modification of the Birman-Schwinger argument, an alternative upper bound on the number of eigenvalues of \(J\). It is also pointed out that this bound is valid for matrix orthogonal polynomials.
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 4098088
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: Green function
zbMATH Keywords: comparison equation
zbMATH Keywords: upper bound on the number of eigenvalues
zbMATH Keywords: Bargmann's argument
zbMATH Keywords: Birman-Schwinger argument
zbMATH Keywords: matrix orthogonal polynomials
author: Jeffrey S. Geronimo
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90022-6
OpenAlex ID: W2020070089
cites work: Discrete and continuous boundary problems
cites work: On the Number of Bound States in a Central Field of Force
cites work: Orthogonal polynomials from the viewpoint of scattering theory
cites work: Q4170993
cites work: Orthogonal Polynomials Whose Distribution Functions Have Finite Point Spectra
cites work: Spectral Properties of Orthogonal Polynomials on Unbounded Sets
cites work: Orthogonal polynomials and measures with finitely many point masses
cites work: Q5786094
cites work: An upper bound on the number of eigenvalues of an infinite dimensional Jacobi matrix
cites work: Scattering Theory and Polynomials Orthogonal on the Real Line
cites work: Q5526485
cites work: Q3215519
cites work: Orthogonal polynomials
cites work: Q5653193
cites work: Q4185363
cites work: Q5667036
cites work: Q5792555
cites work: Q5571161
description: scientific article; zbMATH DE number 4098088