Discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary and transmissions conditions

The purpose of this paper is to study a Sturm-Liouville problem with discontinuities in the case when an eigenparameter appears not only in the differential equation but it also appears in both the boundary and transmission conditions. We suggest a new approach for the definition of a suitable Hilbert space and a symmetric linear operator defined in this space in such a way that the considered problem can be interpreted as the eigenvalue problem of this operator and for construction and approximation of a fundamental solution. We apply these results to find asymptotic formulas of eigenvalues and cot-responding eigenfunctions.