Spectral distribution problem for Sturm-Liouville with operators the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter

In order to enrich the spectral theory of Sturm-Liouvillel (S-L) differential operators, the discontinuous S-L problem with boundary conditions dependent on spectral parameters on closed interval \ is studied. Firstly, by using the equivalent characterization of the problem in the direct sum space, the alternating relation between the eigenvalues of the discontinuous S-L problem and the eigenvalues of the continuous S-L problem is given. That is, there is exactly one eigenvalue of the continuous S-L problem in every open subinterval of the eigenvalues of the discontinuous S-L problem, and then the oscillation theory of the discontinuous S-L problem is derived from the oscillation theory of the continuous S-L problem. Through the transformations of Prüfer and Hergloz function, the transformation between the discontinuous S-L problem with boundary conditions dependent spectral parameters and discontinuous S-L problem with constant boundary conditions is established. The obtained converted eigenvalues are equal to those (excluding the finite eigenvalues) before the conversion. Finally, the asymptotic expressions of eigenvalues of discontinuous S-L problems with boundary conditions dependent on spectral parameters are obtained by constructing the eigenfunctions of discontinuous S-L problems with constant boundary conditions. The new research method can be extended to the study of the S-L problem with boundary conditions dependent spectral parameters