The Spectral Distribution And The Inverse Eigenvalue Problem For The Sturm-Liouville Operators With The Discontinuity Conditions At An Interior Point And Boundary Conditions Depending On The Eigenparameter

Driven by other disciplines and many engineering and technical fields,the study of the Sturm-Liouville problem has aroused great interest and high attention to domestic and overseas scholars.So far,it has become one of the fastest development and growth fields in applied mathematics.The spectral distribution and inverse spectral problems are two basic and important subjects in the problem study,they have extensive and direct applications in the earth physics,the meteorology and other fields,they are also the effective way of solving nonlinear evolution equation in mathematical physics.In this present paper,the spectral distribution and inverse spectral problems are studied for the Sturm-Liouville operators with the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter.We establish the transformations between such problems and asymptotic expressions of eigenvalues.The uniqueness theorems are proved and the reconstruction algorithms are provided by the corresponding spectrum data,respectively.The main works are givenas follows:In the first chapter,we first give a summary of the physical backgrounds of the Sturm-Liouville operators with the discontinuity conditions at an interior point and boundary conditions depending on the eigenpararmeter,and elaborate the re-search advances of the spectral distribution and the inverse eigenvalue problems,then introduce the main work of this paper.In the second chapter,the spectral distribution problem is studied for the Sturm-Liouville operators with the discontinuity conditions at an interior point and boundary conditions depending on the eigenparameter.Using the characterizations of the Sturm-Liouville problem on a direct sum space,we establish the asymp-totic expressions of eigenvalues,oscillation of eigenfunctions,and transformations between such problems.In the third chapter,the inverse spectral problems is studied for the Sturm-Liouville operators with the discontinuity conditions at an interior point and bound-ary conditions depending on the eigenparameter.Using the Weyl function,we pro-vide the uniqueness results for the inverse spectral problems;finally we provide the constructive solutions by the method of spectral mappings.