The weak eigenfunctions of boundary-value problem with symmetric discontinuities

The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions. We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.