On asymptotic stability of standing waves of discrete Schr\"odinger equation in $\Bbb Z$

TL;DR

This paper establishes asymptotic stability of standing waves for the discrete Schrödinger equation on the integer lattice, extending classical results to the discrete setting with less restrictive potential decay conditions.

Contribution

It transposes a continuous Schrödinger stability result to the discrete lattice, relaxing decay conditions on the potential and providing new dispersion estimates.

Findings

Proves asymptotic stability of standing waves on Z.

Establishes decay rate |e^{itH}(n,m)| ≤ C t^{-1/3} under weaker conditions.

Reduces hypotheses on the potential compared to previous works.

Abstract

We prove an analogue of a classical asymptotic stability result of standing waves of the Schr\"odinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi and it involves a discrete Schr\"odinger operator H. The decay rates on the potential are less stringent than in Mizumachi, since we require for the potential $q\in \ell ^{1,1}$. We also prove $|e^{itH}(n,m)|\le C < t > ^{-1/3}$ for a fixed $C$ requiring, in analogy to Goldberg and Schlag only $q\in \ell ^{1,1}$ if $H$ has no resonances and $q\in \ell ^{1,2}$ if it has resonances. In this way we ease the hypotheses on H contained in Pelinovsky and Stefanov, which have a similar dispersion estimate.