Eigenvalues and normalized eigenfunctionsof discontinuous Sturm-Liouville problem with transmission conditions

In this study, discontinuous Sturm-Liouville problems which contain eigenvalue parameters both in equation and in the boundary conditions are investigated. We introduce an operator-theoretic interpretation, extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. By modifying some techniques of [2, 6, 7] we obtain asymptotic formulae for eigenvalues and normalized eigenfunctions. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in [2].