The Basis Property of the System of Weak Eigenfunctions of a Discontinuous Sturm–Liouville Problem

Abstract

The purpose of this study is to investigate some important spectral properties of one discontinuous Sturm–Liouville problem. By applying our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is defined as a concept of weak eigenfunctions. Then we introduce for consideration some compact operators in such a way that this integral equation can be reduced to the appropriate operator-pencil equation and prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. Finally, it is established that the spectrum is discrete and the system of corresponding weak eigenfunctions forms an orthonormal basis of the appropriate Hilbert space.