Asymptotic formulas for eigenfunctions of the Sturm-Liouville problems with eigenvalue parameter in the boundary conditions

In this study, discontinuous Sturm-Liouville problems with eigenvalue parameter in the boundary conditions and with transmission conditions at the point of discontinuity are considered. We construct some basic solutions, derive an asymptotic expression for these solutions and 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. By using these results we find asymptotic formulas for eigenvalues and corresponding eigenfunctions.