On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions

Abstract

Abstract The goal of this study is to analyse the eigenvalues and weak eigenfunctions of a new type of multi-interval Sturm-Liouville problem (MISLP) which differs from the standard Sturm-Liouville problems (SLPs) in that the Strum-Liouville equation is defined on a finite number of non-intersecting subintervals and the boundary conditions are set not only at the endpoints but also at finite number internal points of interaction. For the self-adjoint treatment of the considered MISLP, we introduced some self-adjoint linear operators in such a way that the considered multi-interval SLPs can be interpreted as operator-pencil equation. First, we defined a concept of weak solutions (eigenfunctions) for MISLPs with interface conditions at the common ends of the subintervals. Then, we found some important properties of eigenvalues and corresponding weak eigenfunctions. In particular, we proved that the spectrum is discrete and the system of weak eigenfunctions forms a Riesz basis in appropriate Hilbert space.