DISCONTINUOUS STURM-LIOUVILLE PROBLEMS INVOLVING AN ABSTRACT LINEAR OPERATOR

In this paper we introduce to consideration a new type boundary value problems consisting of an "Sturm-Liouville" equation on two disjoint intervals as $ -p(x)y^{\prime \prime }+ q(x)y+\mathfrak{B}y|_{x} = \mu y , x\in [a, c)\cup(c, b] $ together with two end-point conditions whose coefficients depend linearly on the eigenvalue parameter, and two supplementary so-called transmission conditions, involving linearly left-hand and right-hand values of the solution and its derivatives at point of interaction $x=c, $ where $\mathfrak{B}:L_{2}(a, c)\oplus L_{2}(c, b)\rightarrow L_{2}(a, c)\oplus L_{2}(c, b)$ is an abstract linear operator, non-selfadjoint in general. For self-adjoint realization of the pure differential part of the main problem we define "alternative" inner products in Sobolev spaces, "incorporating" with the boundary-transmission conditions. Then by suggesting an own approaches we establish such properties as topological isomorphism and coercive solvability of the corresponding nonhomogeneous problem and prove compactness of the resolvent operator in these Sobolev spaces. Finally, we prove that the spectrum of the considered eigenvalue problem is discrete and derive asymptotic formulas for the eigenvalues. Note that the obtained results are new even in the case when the equation is not involved an abstract linear operator $\mathfrak{B}.$