Eigenvalue Problems Of Regular Self-Adjoint Sturm-Liouville Operator With Mixed Boundary Condition

In this paper,we investigate the distribution and mutiplicity problems of eigenvalues of the Sturm-Liouville operator with mixed boundary condition.Firstly,we prove that the operator has countable real eigenvalues while b or c≠0 and|δ|< 1,the eigenvalues have no finite cluster point,the sufficiently large eigenvalues are all simple eigenvalues,and we give the asymptotic formula of eigenvalue and eigenvalue function,and furthure,we conclude that the sufficiently large eigenvalues present a kind of uniform distribution different from the situation in S-L problems with seperator boundary condition. Secondly,we study the S-L problems with mixed boundary condition while q(x)= 1 and q(x)= 0,we prove that the eigenvalues are all double eigenvalues while q(x)= 1 andδ=±1,so we solve the left problems in paper [4],and when q(x) = 0,we give a sufficient and necessary condition that the eigenvalues are all double eigenvalues and a reasoning form which we conclude that the eigenvalues of the S-L operator with mixed boundary condition are likely all to be simple eigenvalues.Finally,we study the eigenvalue problem of a kind of discontinous operator with mixed boundary condition.