On discontinuous Sturm-Liouville problems with transmission conditions

We consider a discontinuous Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. By modifying some techniques of [2], [11] and [14] we extend and generalize some approach and results of classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation such a way that the considered problem can be interpreted as an eigenvalue problem of suitable self-adjoint operator, then we construct the Green function and resolvent operator and derive an asymptotic formulas for eigenvalues and normalized eigenfunctions.