Expansion Theorem for Sturm-Liouville problems transmission conditions

TL;DR

This paper extends spectral properties of regular Sturm-Liouville problems to a discontinuous boundary-value problem with transmission conditions, constructing resolvent operators and proving eigenfunction expansion theorems.

Contribution

It introduces an extension of spectral analysis to Sturm-Liouville problems with transmission conditions, including resolvent and Green's function construction.

Findings

Constructed resolvent operator and Green's function.

Proved eigenfunction expansion theorems in modified Hilbert space.

Extended spectral properties to discontinuous Sturm-Liouville problems.

Abstract

The purpose of this paper is to extend some spectral properties of regular Sturm-Liouville problems to the special type discontinuous boundary-value problem, which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions. We construct the resolvent operator and Green's function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space $L_{2}[a, b]$. \vskip0.3cm\noindent Keywords: Boundary-value problems, transmission conditions, Resolvent operator, expansion theorem.