Research On Eigenvalues Of Some Differential Operators

Ordinary differential operators theory can supply differential equations, classical physics, modern physics and other technique fields the theory basis, which is a compositive and edging mathematics branch of ordinary differential equations, functional analysis, space theory, operators theory etc. It contains a great important problems, such as deficiency index theory, adjoint extention, spectral analysis, press eigenfunction to spread, numerical method, inverse questions, and so on.In this paper, we study an important problem in the field of differential operators. Based on book [7] and [10], we first analyze a class of right-definite Sturm-Liouville operators with periodic boundary conditions. With the method of function theory, we solve the problem of existence and distribution of the eigenvalue, prove that the eigenvalue set and zero set of a whole function ofλare same, the rank of eigenvalue and zero multiple number are conincident and the gradually near expression of eigenvalue and eigenfunction.Then the paper studies the total reliability of sort of forth derivation's eigenvalue on the boundary point gets he first order differential equation of eigenvalue to the boundary point under the condition of self-adjoint separated border and discusses some eigenvalue's nature of boundary point function when interval is shrinked to zero.This paper also uses the reliable relation of the eigenvalue of S-L problem on boundary conditions to establish the eigenvalue inequality of left and right rule's S-L problem. At last, the paper researches the eigenvalue inequality of regular sturm-liouwilleproblem.This paper contains four parts. The first part: an introduction of the background of the problems we investigate and main results we obtain in this paper. The second part: a right-definite Sturm-Liouville problem with periodic boundary conditions. The third part: the dependence of the eigenvalue of a class of forth differential operators on boundary point. The fourth part: eigenvalues inequalities.