Selfadjoint realization of boundary-value problems with interior singularities

TL;DR

This paper develops a novel operator-theoretic approach to analyze the spectral properties of Sturm-Liouville problems with interior singularities, establishing selfadjointness and constructing Green's functions.

Contribution

It introduces a new technique for handling interior singularities in Sturm-Liouville problems, including the construction of special solutions and resolvent operators.

Findings

Established selfadjointness of the boundary-value problem.

Derived explicit formulas for Green's functions.

Provided conditions for the existence of solutions.

Abstract

The purpose of this paper is to investigate some spectral properties of Sturm-Liouville type problems with interior singularities. Some of the mathematical aspects necessary for developing own technique presented. By applying this technique we construct some special solutions of the homogeneous equation and present a formula and the existence conditions of Green's function. Further based on this results and introducing operator treatment in adequate Hilbert space we derive the resolvent operator and prove selfadjointness of the considered problem.