On some class of self-adjoint boundary value problems with the spectral parameter in the equations and the boundary conditions

Abstract

The aim of this work is to study the spectral properties of some class of self-adjoint linear boundary value and transmission problems (in domains with Lipschitz boundaries) where equations and boundary conditions depend linearly on the eigenparameter λ. We consider some abstract general problem that can be formulated on the basis of the abstract Green’s formula for a triple of Hilbert spaces and a trace operator. We prove that the spectrum of the abstract general problem consists of real normal eigenvalues with unique limit point ∞, and the system of corresponding eigenelements forms an orthonormal basis in some Hilbert space. We find also some asymptotic formulas for positive or positive and negative branches of eigenvalues. As examples, we consider three multicomponent transmission problems arising in mathematical physics and some abstract general transmission problem.