Oscillation theory and renormalized oscillation theory for Jacobi operators
type: Item
instance of: scholarly article
title: Oscillation theory and renormalized oscillation theory for Jacobi operators
zbMATH Open document ID: 0866.39002
published in: Journal of Differential Equations
publication date: 1997-07-24T00:00:00Z
review text: For the difference operator \[ (Hu)(n)= a(n)u(n+1)- a(n-1)u(n-1)- b(n)u(n) \] the dimension of the spectral projection \(P_{(-\infty,\lambda)}(H)\) is expressed in terms of nodes of any solution of the equation \(Hu=\lambda u\). For example: if \(\lambda\) is an eigenvalue of \(H\), then \[ \text{dim Ran }P_{(-\infty,\lambda)} (H)=\#(u_+(\lambda)), \] where \(\#(u_+(\lambda))\) denotes the total numbers of nodes of the nontrivial solution \(u\) satisfying a boundary condition at \(\infty\). Moreover (reformulating the oscillation theory) the dimension of \(P_{(\lambda_1,\lambda_2)} (H)\) is considered. It is shown that this dimension is equal to the number of nodes of the Wronskian \(W(u_1,u_2)\) of two solutions \(u_1,u_2\) satisfying the equations \(Hu_j= \lambda_ju_j\), \(j=1,2\). As application of the main results the spectra of perturbed periodic Jacobi operators are studied.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A70
zbMATH DE Number: 933284
zbMATH Keywords: renormalized oscillation theory
zbMATH Keywords: difference operator
zbMATH Keywords: spectral projection
zbMATH Keywords: periodic Jacobi operators
author: Gerald Teschl
reviewed by: Jerzy Ryszard Popenda
MaRDI profile type: Publication
OpenAlex ID: W1995475407
Wikidata QID: Q57343973
DOI: 10.1006/JDEQ.1996.0126
description: scientific article; zbMATH DE number 933284