Resolvent Operator and Spectrum of New Type Boundary Value Problems

Abstract

The aim of this study is to investigate a new type boundary value problems which consist of the equation $-y''(x) + (\mathcal{B}y)(x) = \lambda y(x)$ on two disjoint intervals $(-1 , 0) \ \textrm{and} \ (0 , 1)$ together with transmission conditions at the point of interaction $x = 0$ and with eigenparameter dependent boundary conditions, where $\mathcal{B}$ is an abstract linear operator. By suggesting an own approaches we introduce modified Hilbert space and linear operator in it such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. We establish such properties as isomorphism and coerciveness with respect to spectral parameter, maximal decreasing of the resolvent operator and discreteness of the spectrum. Further we examine asymptotic behaviour of the eigenvalues.