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 $\mathbb{Z}$ of a result by Mizumachi [J. Math. Kyoto Univ., 48 (2008), pp. 471–497] and it involves a discrete Schrödinger operator $H=-\Delta+q$. The decay rates on the potential are less stringent than in [J. Math. Kyoto Univ., 48 (2008), pp. 471–497], since we require $q\in\ell^{1,1}$. We also prove $|e^{itH}(n,m)|\le C\langle t\rangle^{-1/3}$ for a fixed C requiring, in analogy to Goldberg and Schlag [Comm. Math. Phys., 251 (2004), pp. 157–178], 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 [On the Spectral Theory and Dispersive Estimates for a Discrete Schrödinger Equation in One Dimension, http://arxiv.org/abs/0804.1963v1], which have a similar dispersion estimate.