On an inverse spectral problem for Sturm–Liouville operator with discontinuous coefficient

Abstract

In this paper, the direct and inverse problems for Sturm–Liouville operator with discontinuous coefficient are studied. The spectral properties of the Sturm–Liouville problem with discontinuous coefficient such as the orthogonality of its eigenfunctions and simplicity of its eigenvalues are investigated. Asymptotic formulas for eigenvalues and eigenfunctions of this problem are examined. The resolvent operator is constructed and the expansion formula with respect to eigenfunctions is obtained. It is shown that eigenfunctions of this problem are in the form of a complete system. The Weyl solution and Weyl function are defined. Uniqueness theorems for the solution of the inverse problem according to Weyl function and spectral date are proved.