Completeness of eigenfunctions of Sturm-Liouville problems with discontinuities at three points

TL;DR

This paper investigates Sturm-Liouville problems with discontinuities at three points, establishing the completeness of eigenfunctions and analyzing eigenvalue properties within a self-adjoint operator framework.

Contribution

It introduces a framework for analyzing discontinuous Sturm-Liouville problems with transmission conditions at three points, proving eigenfunction completeness and eigenvalue simplicity.

Findings

Eigenfunctions are complete in the Hilbert space.

Eigenvalues are analytically simple.

The problem is modeled via a self-adjoint operator.

Abstract

In this work, we study discontinuous Sturm-Liouville type problems with eigenparameter dependent boundary condition and transmission conditions at three interior points. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We show that the eigenvalues of the problem are analytically simple, and the eigenfunctions of A are complete in H.