On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm-Liouville problem
type: Item
instance of: scholarly article
title: On a computer assisted proof of the existence of eigenvalues below the essential spectrum of the Sturm-Liouville problem
zbMATH Open document ID: 0971.65069
DOI: 10.1016/S0377-0427(00)00481-7
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 2001-11-06T00:00:00Z
review text: A class of self-adjoint Sturm-Liouville eigenvalue problems on \([0,\infty)\) defined by \(-y''+qy=\lambda y\), \(y(0)=0\), where \(q\) is subject to certain assumptions, is considered. In an earlier paper [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No.~1986, 2229-2234 (1999; Zbl 0934.34073)] the authors have presented a new method for proving the existence of eigenvalues below the essential spectrum. This method combines operator theory and numerical analysis including interval analysis. In this paper, the authors develop further the method to cover a greater class of problems. For various values of the positive constant \(c\), eigenvalues below the essential spectrum are calculated for \(q(x)=-c \text{ exp}(-x/4)\cos(x)\), \(q(x) = -c \text{exp}(- x^2)\), and \(q(x) = c \sin(x + 1/(1 + x^2))\).
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 34L16
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65G40
Mathematics Subject Classification ID: 68T15
zbMATH DE Number: 1566025
zbMATH Keywords: self-adjoint Sturm-Liouville eigenvalue problems
zbMATH Keywords: existence of eigenvalues below the essential spectrum
zbMATH Keywords: interval analysis
reviewed by: Waldemar Velte
MaRDI profile type: Publication
cites work: Q3345690
cites work: Regular approximations of singular Sturm-Liouville problems
cites work: Spectral concentration and rapidly decaying potentials
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cites work: Safe numerical bounds for the Titchmarsh–Weyl m(λ)-function
cites work: On the existence of an eigenvalue below the essential spectrum
cites work: Q4774804
cites work: Q3813179
cites work: Spectral theory of ordinary differential operators
Wikidata QID: Q127436369
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description: scientific article; zbMATH DE number 1566025