The Inverse Spectral Problems Of Sturm-Liouville Problems With Eigenvalue Parameter Contained In The Boundary Conditions

Inverse Sturm-Liouville problem has been an important basis curriculum in applied mathematics, and it has a broad and immediate application in the field of geophysics, quantum mechanics. In this thesis, we study the direct and inverse eigenvalue problems for two kind of Sturm-Liouville differential operators, one of them is discontinuous problems with internal interface conditions and the other is problems with eigenvalue parameter contained in the boundary conditions. The main works of the paper are the following.In the first chapter, we introduce the background and present situation of in-verse Sturm-Liouville problems, and summarize the main conclusions of this article.In the second chapter, consider the inverse eigenvalue problem for discontinuous Sturm-Liouville problem. When the set of the eigenvalues of a Sturm-Liouville problem is known, and a finite number of eigenvalues of another problem by changing one boundary condition of the further one is unknown, we give conditions for their potential function, which is a generalization of Hochstadt’s theorem for the classic Sturm-Liouville problem.In the third chapter, consider the direct problem for the Sturm-Liouville prob-lem with eigenvalue parameter contained in the boundary conditions. We give the asymptotic formulas of the eigenvalues and eigenfunctions of this problem and dis-cuss the expansion problem via its eigenfunctions.In the fourth chapter, consider the inverse problems for the Sturm-Liouville problem with eigenvalue parameter contained in the boundary conditions. We obtain conditions for their potential function, when the spectrum a problem is known, and a finite number of eigenvalues of another problem by changing one boundary condition of the further one is unknown.