A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry

The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with thei...

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Main Author: S. B. Rutkevich
Format: Article
Language:English
Published: Wiley 2018-01-01
Series:Advances in Mathematical Physics
Online Access:http://dx.doi.org/10.1155/2018/9784091
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author S. B. Rutkevich
author_facet S. B. Rutkevich
author_sort S. B. Rutkevich
collection DOAJ
description The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.
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spelling doaj-art-e0e13472219644a49f778b112920010c2025-02-03T06:46:26ZengWileyAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/97840919784091A Formula for Eigenvalues of Jacobi Matrices with a Reflection SymmetryS. B. Rutkevich0Fakultät für Physik, Universität Duisburg-Essen, Duisburg 47048, GermanyThe spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the 2M-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new polynomial identity relating the eigenvalues of such matrices with their matrix entries is obtained. In the limit M→∞ this identity induces some requirements, which should satisfy the scattering data of the resulting infinite-dimensional Jacobi operator in the half-line, of which super- and subdiagonal matrix elements are equal to -1. We obtain such requirements in the simplest case of the discrete Schrödinger operator acting in l2(N), which does not have bound and semibound states and whose potential has a compact support.http://dx.doi.org/10.1155/2018/9784091
spellingShingle S. B. Rutkevich
A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
Advances in Mathematical Physics
title A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
title_full A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
title_fullStr A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
title_full_unstemmed A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
title_short A Formula for Eigenvalues of Jacobi Matrices with a Reflection Symmetry
title_sort formula for eigenvalues of jacobi matrices with a reflection symmetry
url http://dx.doi.org/10.1155/2018/9784091
work_keys_str_mv AT sbrutkevich aformulaforeigenvaluesofjacobimatriceswithareflectionsymmetry
AT sbrutkevich formulaforeigenvaluesofjacobimatriceswithareflectionsymmetry