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Eigenvalue asymptotics for the sturm-liouville operator with potential having a strong local negative singularity

Journal article
Authors Medet Nursultanov
Grigori Rozenblioum
Published in Opuscula Mathematica
Volume 37
Pages 109-139
ISSN 1232-9274
Publication year 2017
Published at Department of Mathematical Sciences
Pages 109-139
Language en
Links dx.doi.org/10.7494/OpMath.2017.37.1...
Keywords Asymptotics of eigenvalues, Singular potential, Sturm-Liouville operator
Subject categories Mathematics

Abstract

© Wydawnictwa AGH, Krakow 2017.We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.

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