Continuous dependence of the n-th eigenvalue of self-adjoint discrete Sturm–Liouville problems on the problem

Abstract

This paper is concerned with continuous dependence of the n-th eigenvalue of self-adjoint discrete Sturm–Liouville problems on the problem. 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 and several properties of the n-th eigenvalue function on a subset 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 problem. Continuous dependence of the n-th eigenvalue function on the equation and on the boundary condition is studied separately. Consequently, the continuity and discontinuity of the n-th eigenvalue function are completely characterized on the whole space of the problems. Especially, asymptotic behavior of the n-th eigenvalue function near each discontinuity point is given.