Dependence Of Eigenvalues Of Sturm-Liouville Problems On The Coefficients

In this thesis,we first study the dependence of eigenvalues of regular dis-crete Sturm-Liouville problems on the coefficients of the problems.A regular discrete Sturm-Liouville problem consists of a regular discrete Sturm-Liouville equation and a boundary condition.We know that the eigenvalues vary as the coefficients in the equation and in the boundary condition change.So,how the eigenvalues vary as the coefficients change is what we want to study.For ex-ample,whether the eigenvalue is continuously dependent on the coefficients;whether the eigenvalue changes in a monotonic way as some parameter in the coefficients varies.We then in this thesis study the dependence of the n-th eigenvalue of the regular self-adjoint discrete Sturm-Liouville problems on the coefficients in the equations and in the boundary conditions.In general,the n-th eigen-value does not continuously depend on the coefficients in the equations and in the boundary conditions.How to find all such discontinuous coefficients in the equations and in the boundary conditions,and what is the asymptotic behavior of the n-th eigenvalue near the discontinuity point are what we also want to study.We find that both the method and the results of the research on the dependence of the n-th eigenvalue on the coefficients of the regular discrete Sturm-Liouville problems are quite different from that in the regular continuous Sturm-Liouville problems.Moreover,with the development of the computer technology,the research on the numerical computing has been exten-sively attractive.Our study forms a theoretical foundation for the numerical works on discrete Sturm-Liouville problems.We also study the dependence of isolated eigenvalues on the coefficients in the boundary conditions of self-adjoint continuous Sturm-Liouville problems with one singular endpoint.Since the problem is a singular one,it makes dif-ficulties.In particular,when the singular endpoint is in the limit point case,the method that used in the dependence of isolated eigenvalues of regular self-adjoint Sturm-Liouville problem does not work here.The Weyl-Titchmaxsh m(A)-function theory and its relationship with the spectrum of the singular problems is an important tool to study this problem.The a priori estimate of the bound of solutions also plays an important role in studying this problem.The research on this topic has important applications in physics.For exam-ple,the research on the linearized stability and instability of some flow,the integrability of a model for shallow water waves,which is the Camassa-Holm equation.Perturbation theory of eigenvalues of regular self-adjoint continuous Sturm-Liouville problems has been extensively investigated.For the dependence of eigenvalue of regular self-adjoint continuous Sturm-Liouville problems on the coefficients of the problems,there are many results.Kong,Wu,and Zettl showed the continuous dependence of the eigenvalues on the equations and the boundary conditions,formed the continuous eigenvalue branches,and gave the derivative formulas of the continuous eigenvalue branches[35,37].Fur-ther,they showed that the n-th eigenvalue function is continuously dependent on the continuous Sturm-Liouville equations in general,and is not continuous-ly dependent on the boundary conditions.They also completely characterized the discontinuity of the n-th eigenvalue as a function defined on the bound-ary conditions[33].For the general regular Sturm-Liouville problems on time scales,Kong showed that the n-th eigenvalue depends continuously on the separated boundary conditions except at some special ones[32].For the sin-gular continuous Sturm-Liouville problems,Zhang,Sun,and Zettl showed the n-th eigenvalue below the lower bound of the essential spectrum is continuous-ly dependent on all the parameters in the boundary conditions and gave its derivative formulas[57].The following is the organization of this thesis.This thesis is divided into five chapters.In Chapter 1,basic concepts and preliminary results are introduced,including the space of regular discrete Sturm-Liouville equations,the space of boundary conditions,the space of reg-ular discrete Sturm-Liouville problems,the space of regular self-adjoint dis-crete Sturm-Liouville problems,the basic properties of eigenvalues of regular discrete Sturm-Liouville problems,the space of boundary conditions of self-adjoint continuous Sturm-Liouville problems with one singular endpoint,and the basic properties of eigenvalues of self-adjoint continuous Sturm-Liouville problems with one singular endpoint.In Chapter 2,the dependence of eigenvalues of regular discrete Sturm-Liouville problems on the coefficients of the problems is studied.Relationships between the analytic and geometric multiplicities of an eigenvalue of a given Sturm-Liouville problem are given and a direct proof is given to show that for a self-adjoint Sturm-Liouville problem,the two multiplicities of an eigenvalue coincide.It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem.So,all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the s-pace of problems,and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems.The analyticity,differentiability and monotonicity of continuous eigenvalue branches are further studied.In Chapter 3,continuous dependence of the n-th eigenvalue of regular self-adjoint discrete Sturm-Liouville problems on the problem is studied.The n-th eigenvalue is considered as a function on the space of the problems,called the n-th eigenvalue function.A necessary and sufficient condition for all the n-th eigenvalue functions to be continuous on a connected subset of the space of the problems is that the numbers of eigenvalues for all Sturm-Liouville problems in the subset are equal.Several properties of the n-th eigenvalue function on a subset of the space of the problems are given,including that monotonicity of the n-th eigenvalue function in some parameter yields its asymptotic behavior along the direction of the parameter near the discontinuity point.They play an important role in the study of continuous dependence of the n-th eigenvalue function on the problems.The continuity sets and discontinuity sets of the n-th eigenvalue function on the space of the equations,boundary conditions,and problems are given and its asymptotic behavior near a discontinuity point is also presented.In Chapter 4,continuous and differentiable dependence of isolated eigen-values on the boundary conditions of self-adjoint continuous Sturm-Liouville problems with one singular endpoint is studied.Locally continuous dependence of eigenvalues on the boundary conditions is proved.If the endpoint b is in the limit circle case,we shall approach a similar method used in the regular prob-lems[35,37]to prove the locally continuous dependence of eigenvalues on the singular problems.If the endpoint b is in the limit point case,the spectrum of the considered problem may contain essential spectrum,which causes some d-ifficulties.The method mentioned above does not work here.To overcome the difficulties,we shall employ the Weyl-Titchmarsh m(?)-function in a neigh-borhood of an isolated eigenvalue and its relationships with the spectrum of the singular problems to obtain the expected results.Then continuous eigen-value branches through each isolated eigenvalue over the space of boundary conditions are formed.It is rigorously shown that the eigenfunctions for the eigenvalues along a continuous simple eigenvalue branch can be continuously chosen with uniform bound inw2 norm,which enables us to take limits under improper integration in computing the derivative of the continuous eigenvalue branch.The derivative formulas of the continuous simple eigenvalue branch with respect to all the parameters in the boundary conditions are given,and thus its monotonicity with respect to some parameters is derived.In Chapter 5,conclusion of the thesis and prospect for future works are given.