On the Boundary Value Problem for Discontinuous Sturm–Liouville Operator

Abstract

In this paper, we study the Sturm–Liouville problem which has not only the discontinuous coefficient, but also the discontinuity condition at an interior point of a finite interval. The new integral representation for the solution of the discontinuous Sturm–Liouville equation is constructed and the properties of the its kernel function are given. The asymptotic formulas of the eigenvalues and eigenfunctions of this problem are examined. The completeness theorem and expansion theorem of eigenfunctions are proved.