Completeness of eigenfunctions of discontinuous Sturm-Liouville problems

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] circle plus C and suitable self-adjoint linear operator in it