Resolvent Operator and Eigenfunctions of a Sturm-Liouville Type problem

TL;DR

This paper studies the resolvent operator and eigenfunction completeness for a Sturm-Liouville problem with boundary discontinuities and an eigenparameter, using operator theory in a specially defined Hilbert space.

Contribution

It introduces an operator-theoretic framework for a Sturm-Liouville problem with boundary discontinuities and an eigenparameter, establishing self-adjointness and spectral properties.

Findings

Resolved the operator-theoretic formulation of the problem.

Proved the completeness of eigenfunctions.

Established the self-adjointness of the associated operator.

Abstract

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm-Liouville problem with discontinuities at two points. The problem contains an eigenparameter in the one of boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space and suitable self-adjoint lineer operator in it.