Hierarchies of difference boundary value problems II – Applications

Abstract

This paper provides an illustration of the work done in [14] where a hierarchy of difference boundary value problems was developed. In particular, we studied the effect of applying a Crum-type transformation to a weighted second order difference equation with general λ-dependent boundary conditions at the end points, for eigenparameter λ. In this paper we demonstrate by means of examples, how it is possible to go up and down the hierarchy of boundary value problems developed in [14], keeping the form of the second order difference equation constant but possibly increasing or decreasing the dependence on λ of the boundary conditions at each step. In addition, the transformed boundary value problem either gains or loses an eigenvalue or the number of eigenvalues remains the same as we step up or down the hierarchy.