completeness of eigenfunctions of a discontinuous sturm-liouville problem with eigenparameter-dependent boundary conditions

In this paper, we investigate a class of sturm-liouville problems with eigenparameter dependent boundary conditions and transmission conditions at an interior point. a self-adjoint linear operator is defined in a suitable hilbert space such that the eigenvalues of such a problem coincide with those of then operator. then we study the resolvent operator and completeness of eigefunctions of the sturm-liouville problem and we derive the main result about expansion in series of eigenfunctions.