ON ASYMPTOTIC STABILITY OF STANDING WAVES OF DISCRETE SCHRODINGER EQUATION IN Z

We prove an analogue of a classical asymptotic stability result of standing waves of the Schrodinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi [J. Math. Kyoto Univ., 48 (2008), pp. 471-497] and it involves a discrete Schrodinger 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 is an element of l(1,1). We also prove vertical bar e(itH)(n, m)vertical bar <= C < t >(-1/3) for a fixed C requiring, in analogy to Goldberg and Schlag [Comm. Math. Phys., 251 (2004), pp. 157-178], only q is an element of l(1,1) if H has no resonances and q is an element of l(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 Schrodinger Equation in One Dimension, http://arxiv.org/abs/0804.1963v1], which have a similar dispersion estimate.