A Singular Sturm-Liouville Problem with Limit Circle Endpoints and Eigenparameter Dependent Boundary Conditions

Jinming Cai,Zhaowen Zheng

doi:10.1155/2017/9673846

Abstract

In this paper, we investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions. Operator formulation is constructed and asymptotic formulas for eigenvalues and fundamental solutions are given. Moreover, the completeness of eigenfunctions is discussed.

Highlights

We investigate a class of discontinuous singular Sturm-Liouville problems with limit circle endpoints and eigenparameter dependent boundary conditions
It is well known that many topics in mathematical physics require the investigation of eigenvalues and eigenfunctions of the Sturm-Liouville problems
The regular Sturm-Liouville problems with transmission conditions containing an eigenparameter on one of the boundary conditions have received a lot of attention in research

Summary

Introduction

It is well known that many topics in mathematical physics require the investigation of eigenvalues and eigenfunctions of the Sturm-Liouville problems. Some researchers studied the regular Sturm-Liouville problems with eigenparameter on both of the boundary conditions (see [23,24,25]) In these papers, Yang and Wang in [18] considered a Sturm-Liouville problem with discontinuities at two points and eigenparameter dependent boundary condition at one endpoint; they obtained the fundamental solutions and gave the asymptotic formulas of eigenvalues and fundamental solutions. We will consider the following singular discontinuous Sturm-Liouville problem with two limit-circle endpoints and eigenparameter in the boundary conditions: Ly fl − (p (x) y󸀠 (x))󸀠 + q (x) y (x) = λy (x) , (1). G, α, β ∈ D(L), while D(L) is defined in Lemma 1, one has

Operator Formulation

Asymptotic Approximation of Fundamental Solutions

Asymptotic Formulas for Eigenvalues and Eigenfunctions

Completeness of Eigenfunctions