Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition

Khanlar R Mamedov,F Ayca Cetinkaya

doi:10.1186/1687-2770-2013-183

Khanlar R Mamedov, F Ayca Cetinkaya

Open Access

https://doi.org/10.1186/1687-2770-2013-183

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Journal: Boundary Value Problems	Publication Date: Aug 14, 2013
Citations: 19	License type: CC BY 2.0

Affiliation: Mersin University

Abstract

This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.

Highlights

In recent years, there has been a growing interest in physical applications of boundary value problems with a spectral parameter, contained in the boundary conditions
The inverse problem has been analyzed by zeros of the eigenfunctions in [ ], by numerical methods in [ ] and by two spectra, consisting of sequences of eigenvalues and the normed constants in [ ]
In [, ], eigenvalue-dependent inverse problem with the discontinuities inside the interval was examined by the Weyl function

Summary

Introduction

There has been a growing interest in physical applications of boundary value problems with a spectral parameter, contained in the boundary conditions. The relationship between diffusion processes and Sturm-Liouville problem with eigen-parameter in the boundary conditions has been shown in [ ]. Another example of this relationship between the same problem and the wave equation has been examined in [ , ]. Inverse problems for the Sturm-Liouville operator with spectral parameter, contained in the boundary conditions, have been investigated, and the uniqueness of the solution of these problems has been shown in [ – ]. In [ , ], eigenvalue-dependent inverse problem with the discontinuities inside the interval was examined by the Weyl function. Discontinuous and no eigenvalue parameter containing direct problem and inverse problem with the Weyl function were discussed in [ , ]. The similar problem was investigated in the half line by scattering data in [ , ]

We consider the boundary value problem

Let us define

Proof Since

Let us denote that