Hilbert Space Method for Sturm-Liouville Problems with Discontinuities

Boundary-value problems with eigenparameter dependent boundary conditions is one of the most extensively developing fields in pure and applied mathematics. First, we cite the works of Walter and Fulton, both of which have extensively bibliographies, a discussion of physical applications. Walter had given an Hilbert space formulation of such type eigenvalue problems and obtained the expansion result using the selfadjointness of the operator associated with the Sturm-Liouville problem. In recent years there has been increasing interest of Sturm-Liouville type problems with supplementary transmission conditions. For example, the first author of this study and some others have developed the spectral theory of discontinuous Sturm-Liouville problems with additional transmission conditions at the interior points. The concept of generalized solutions in a Hilbert space allows the eigenvalue problem to be reduced to an operator-pencil equation. In this study, we prove that the weak eigenfunctions of the Sturm-Liouville problem with additional transmission conditions form a Riesz basis of the corresponding Hilbert space. First, the generalized solution of the Sturm-Liouville problem is defined as an element of a direct sum spaces with satisfies some integral equalities. Second, using the Riesz representation theorem these equalities are reduced to an operator-pencil equation. Finally, it is established that the eigenfunctions of the original boundary-value-transmission problem form a Riesz basis of suitable Hilbert space.