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

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

2011-09-22

arxiv.org/abs/1109.4691v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

On the behavior at infinity of solutions to difference equations in Schroedinger form does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the behavior at infinity of solutions to difference equations in Schroedinger form, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the behavior at infinity of solutions to difference equations in Schroedinger form will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-262097