On the behavior at infinity of solutions to difference equations in Schroedinger form

TL;DR

This paper investigates the asymptotic behavior of solutions to discrete Schrödinger equations, introducing new methods such as a discrete Liouville-Green transformation and analyzing the structure of transfer matrices.

Contribution

It develops a sharp discrete analogue of the Liouville-Green transformation and explores the behavior at infinity of solutions, including exponential decay and the structure of transfer matrices.

Findings

Perturbed systems share the same asymptotic behavior as original systems.

Established exponential dichotomy and transfer matrix structure.

Derived conditions for exponential decay of solutions using an Agmon metric.

Abstract

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices. Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. A discussion of the discrete Schroedinger problem and its connection with orthogonal polynomials on the real line is presented in an Appendix.