In this study, we consider a new type boundary value problem consisting of a Sturm-Liouville equation on two disjoint intervals together with interaction conditions and with eigenvalue parameter in the boundary conditions. We suggest a special technique to reduce the considered problem into an integral equation by the use of which we define a new concept, the so-called weak eigenfunction for the considered problem. Then we construct some Hilbert spaces and define some self-adjoint compact operators in these spaces in such a way that the considered problem can be interpreted as a self-adjoint operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of weak eigenfunctions form a Riesz basis of the suitable Hilbert space.

The aim of this study is to investigate a new type boundary value problems which consist of the equation −y (x) + (By)(x) = λy(x) on two disjoint intervals (−1, 0) and (0, 1) together with transmission conditions at the point of interaction x = 0 and with eigenparameter dependent boundary conditions, where B is an abstract linear operator, unbounded in general, in the direct sum of Lebesgue spaces L 2 (−1, 0)⊕ L 2 (0, 1). 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.

We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piecewise continuous potential together with eigenparameter dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular, it is shown that the problem under consideration has precisely denumerable many eigenvalues λ 1 , λ 2 , ..., which are real and tends to +∞. Moreover, it is proven that the generalized eigenvectors form a Riesz basis of the adequate Hilbert space.

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We investigate a new type boundary value problem consisting of a differential-operator equation, eigendependent boundary conditions, and two supplementary conditions so-called interface conditions. We give a characterisation of some spectral properties of the considered problem. Particularly, it is established such properties as isomorphism and coerciveness, discreteness of the spectrum and found asymptotic formulas for eigenvalues.