A Summary Of Eigenvalue Problem Of Sturm-Liouville Systems

The problem of Sturm-Liouville as the fundation of the definite problem of equations of mathematical physics such as wave equation, heat conduction equation and so on have been widely used in mathematical physics, physical geography, quantum mechanics and many other fields. Especially in quantum mechanics, the problem of Sturm-Liouville is a basic method of describing microcosmic particle state.We usually convert problems into the eigenvalue and eigenfunction prob-lem of Sturm-Liouville systems by variable separation method in solving free vibration of string and heat conduction problems of thin rod.we usually name the second order ordinary differential equation like (P(x)u’(x))’-Q(x)u(x)+ λρ(x)u(x)= 0, x ∈ [a, b] Sturm-Liouville equations. We get the eigenvalue problem of Sturm-Liouville systems if we attach the equation of some homogeneous boundary conditions.In this paper, we summarized some conclusions of eigenvalue problem of Sturm-Liouville systems, such as the asymptotic behavior of solutions and estimation of eigenvalues with a small perturbation, Hill-type formula and trace formula of different boundary conditions. Furthermore, we get a series of identical equations by trace formula.The article is organized as follows:The first chapter introduces the re-search background of the problem of Sturm-Liouville and the background of Hill-type formula and trace formula; The second chapter summarized the asymptotic behavior of solutions on the conditions of the solution or inde-pendent variable is very large, estimation of eigenvalues and eigenvector with a small perturbation; We introduce the Hill-type formula and trace formula separately on the condition of periodic boundary and separated boundary; In the fifth chapter we use the trace formula of Sturm-Liouville system get a series of identical equations.