Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function

TL;DR

This paper extends spectral property analysis to Sturm-Liouville problems with discontinuous weights at interior points, providing asymptotic formulas for eigenvalues and eigenfunctions using operator theory.

Contribution

It introduces an operator-theoretic approach to analyze Sturm-Liouville problems with discontinuous weights and derives asymptotic formulas for their eigenvalues and eigenfunctions.

Findings

Extended spectral properties to discontinuous weight problems

Derived asymptotic formulas for eigenvalues

Analyzed eigenfunctions' behavior asymptotically

Abstract

In this paper, by using the similar methods of [O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694] we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. We give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions.