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arXiv · 0808.2024

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

Scipio Cuccagna·Mirko Tarulli
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Abstract

We prove an analogue of a classical asymptotic stability result of standing waves of the Schrödinger 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 Schrödinger 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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Scipio Cuccagna, Mirko Tarulli. 2009-03-25. On asymptotic stability of standing waves of discrete Schrödinger equation in $\Bbb Z$. https://arxiv.org/abs/0808.2024

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