Sturm Liouville Problem with Moving Discontinuity Points

TL;DR

This paper introduces a novel Sturm Liouville problem with symmetric moving discontinuities, analyzing its eigenvalues, eigenfunctions, and Green's function, with implications for differential equations with complex boundary and transmission conditions.

Contribution

It presents a new Sturm Liouville problem featuring moving discontinuities, eigenparameter in boundary conditions, and coupled transmission conditions, along with eigenvalue asymptotics and Green's function construction.

Findings

Eigenvalues and eigenfunctions characterized asymptotically.

Green's function constructed for the problem.

Properties of eigenvalues analyzed with new discontinuity features.

Abstract

In this paper, we present a new discontinuous Sturm Liouville problem with symmetrically located discontinuities which are defined depending on a neighborhood of a midpoint of the interval. Also the problem contains an eigenparameter in one of the boundary conditions and has coupled transmission conditions at the discontinuity points. We investigate the properties of the eigenvalues, obtain asymptotic formulas for the eigenvalues and the corresponding eigenfunctions and construct Green's function of this problem.