Hierarchies of Difference Boundary Value Problems
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Boundary Value Problems 2011, 2011:743135 doi:10.1155/2011/743135Published: 18 January 2011
This paper generalises the work done in Currie and Love (2010), where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with various combinations of Dirichlet, non-Dirichlet, and affine -dependent boundary conditions at the end points, where is the eigenparameter. We now consider general -dependent boundary conditions. In particular we show, using one of the Crum-type transformations, that it is possible to go up and down a hierarchy of boundary value problems keeping the form of the second-order difference equation constant but possibly increasing or decreasing the dependence on of the boundary conditions at each step. In addition, we show that 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.