Darboux transformations and second order difference equations
arXiv:1807.06895
Abstract
In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schrödinger equation. The technique is based on the use of first order operators intertwining two difference operators of second order. This method, which has been applied successfully for differential cases, leads also to interesting non trivial results in the discrete case. The technique allows us to construct the solutions for a wide class of difference Schrödinger equations. The exact solutions for some special potentials are also found explicitly.
21 pages