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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.

This article considers a Sturm–Liouville-type problem on a finite number disjoint intervals together with transmission conditions at the points of interaction. We introduce a new operator-theoretic formulation in such a way that the problem under consideration can be interpreted as a spectral problem for a suitable self-adjoint operator. We investigate some principal properties of eigenvalues, eigenfunctions, and resolvent operator. Particularly, by applying Green’s function method, it is shown that the problem has only point spectrum, and the set of eigenfunctions form a basis of the adequate Hilbert space.

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The purpose of this study is to investigate a new class of boundary value transmission problems (BVTPs) for a Sturm-Liouville equation on two separate intervals. We introduce a modified inner product in the direct sum space L 2 [a, c) ⊕ L 2 (c, b] ⊕ C 2 and define a symmetric linear operator in it in such a way that the considered problem can be interpreted as an eigenvalue problem of this operator. Then, by suggesting own approaches, we construct the Green's function for the BVTP under consideration and find the resolvent function for the corresponding inhomogeneous problem.