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

- Series
- Analysis Seminar
- Time
- Wednesday, January 18, 2012 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Lillian Wong – Georgia Tech

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.This talk is based on joint work with Evans Harrell.