Continuous Dependence of the n-th Eigenvalue on Self-adjoint Discrete Sturm-Liouville Problem

Abstract

This paper is concerned with continuous dependence of the n-th eigenvalue on self-adjoint discrete Sturm-Liouville problems. The n-th eigenvalue is considered as a function in the space of the problems. A necessary and sufficient condition for all the eigenvalue functions to be continuous and several properties of the eigenvalue functions in a set of the space of the problems are given. They play an important role in the study of continuous dependence of the n-th eigenvalue function on the problems. Continuous dependence of the n-th eigenvalue function on the equations and on the boundary conditions is studied separately. Consequently, the continuity and discontinuity of the n-th eigenvalue function are completely characterized in the whole space of the problems. Especially, asymptotic behaviors of the n-th eigenvalue function near each discontinuity point are given.