Completeness of eigenfunctions of discontinuous Sturm–Liouville problems

Abstract

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm–Liouville problem with discontinuities at two points. The problem contains an eigenparameter in one of the boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space \( L_{2} [ - 1,1] \oplus C \) and suitable self-adjoint linear operator in it.