Riesz Basis Property of Weak Eigenfunctions for Boundary-Value Problem with Discontinuities at Two Interior Points

Abstract

We investigate one discontinuous boundary value problem which consist of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter-dependent boundary conditions and four supplementary transmission conditions. We establish some spectral properties of the considered problem. For the problem under consideration we define a new concept so-called weak eigenfunctions which is an extension of a classical eigenfunction and prove that the system of weak eigenfunctions form a Riesz basis of the appropriate Hilbert space for the modified Lebesgue space.