On asymptotic stability of standing waves of discrete Schrödinger equation in $\mathbb{Z}$

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.

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

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Cited by in corpus (4)

• Internal modes of discrete solitons near the anti-continuum limit of the dNLS equation
• Asymptotic stability of small gap solitons in the nonlinear Dirac equations
• Orbitally but not asymptotically stable ground states for the discrete NLS
• Dispersive estimate for quasi-periodic Schrödinger operators on 1-$d$ lattices