Some spectral properties of one Sturm–Liouville type problem with discontinuous weight

We consider a discontinuous weight Sturm–Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. We extend and generalize some approaches and results of the classic regular Sturm–Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation in such a way that the problem under consideration can be interpreted as an eigenvalue problem for a suitable selfadjoint operator, construct the Green's function and resolvent operator, and derive asymptotic formulas for eigenvalues and normalized eigenfunctions.