Completeness of one two-interval boundary value problem with transmission conditions

TL;DR

This paper introduces a new integral equation approach to analyze the completeness of eigenfunction expansions for a two-interval Sturm-Liouville problem with transmission conditions, using Green's function in a modified Hilbert space.

Contribution

It develops a novel method combining integral equations and Green's function techniques for spectral analysis of Sturm-Liouville problems with transmission conditions.

Findings

Establishes completeness of eigenfunction expansions under transmission conditions

Develops Green's function approach in a modified Hilbert space

Provides spectral analysis framework for two-interval Sturm-Liouville problems

Abstract

This paper presents a new approach to the two-interval Sturm-Liouville eigenfunction expansions, based essentially on the method of integral equations. We consider the Sturm-Liouville problem together with two supplementary transmission conditions at one interior point. Further we develop Green's function method for spectral analysis of the considered problem in modified Hilbert space.