  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.
