. A P ] 2 8 A ug 2 00 8 ON ASYMPTOTIC STABILITY OF STANDING WAVES OF DISCRETE SCHRÖDINGER EQUATION IN Z

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 [M1] and it involves a discrete Schrödinger operator H = −∆+q. The decay rates on the potential are less stringent than in [M1], since we require q ∈ l1,1. We also prove |eitH(n,m)| ≤ C〈t〉−1/3 for a fixed C requiring, in analogy to Goldberg & Schlag [GSc], only q ∈ l1,1 if H has no resonances and q ∈ l1,2 if it has resonances. In this way we ease the hypotheses on H contained in Pelinovsky & Stefanov [PS], which have a similar dispersion estimate. §