On asymptotic stability of standing waves of discrete Schrödinger equation in $\Bbb Z$

Details On asymptotic stability of standing waves of discrete Schrödinger equation in $\Bbb Z$ On asymptotic stability of standing waves of discrete Schrödinger equation in $\Bbb Z$

2008-08-14

arxiv.org/abs/0808.2024v6

Mathematics

Analysis of PDEs

This is the revised version, to appear on SIAM Jornal of mathematical Analysis

Scientific paper

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.

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