Completeness of Eigenfunctions of Sturm-Liouville Problems with Transmission Conditions

Abstract

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 $A$ is defined in a suitable Hilbert space $H $such that the eigenvalues of such a problem coincide with those of $A$. We show that the operator $A$ has only point spectrum, the eigenvalues of the problem are algebraically simple, and the eigenfunctions of $A$ are complete in $H$.