Study On The Sturm-Liouville Eigenvalue Problems

It is of great significance to study Sturm-Liouville eigenvalue problems from the perspective of mathematics,physics and other disciplines.The continuity of Sturm-Liouville eigenvalues depends on the boundary conditions,and the boundary conditions can be expressed as the direct sum of the Lagrangian space in the phase space of the starting point and the end point.This thesis summarizes the continuous dependence of the j-th eigenvalue of the Sturm-Liouville problems on the Lagrangian boundary conditions in one dimension and higher dimensions.On the premise of"layering" the Lagrangian-Grassmann manifold according to the dimension of the intersection with Dirichlet boundary conditions,we get the results that the j-th eigenvalue is continuous"within the layer" and discontinuous"between the layers".Based on the continuity,the layer range of the j-th eigenvalue in each layer is obtained.Finally,we use the Morse index theorem of Lagrangian system to give a new proof of the layer range.This thesis is divided into six chapters:the first chapter introduces the background of Sturm-Liouville eigenvalue problems;The second chapter gives the preliminaries needed in this paper,including the basic knowledge of complex symplectic geometry,spectral theory of selfadjoint operators and Sobolev embedding theorem.In Chapter 3,the continuity and the range of one-dimensional Sturm-Liouville eigenvalue problems are introduced,and several examples are given.The fourth chapter introduces the index theory of Lagrangian subspaces,including Maslov index,Hormander index and triple index.In Chapter 5,the correlation results of the continuity and the layer range of higher-dimensional Sturm-Liouville eigenvalues are given.In Chapter 6,the Morse index theorem of Lagrangian systems is introduced and its application to Sturm-Liouville eigenvalue problems in higher dimensions is given.