Dynamical localization II with an application to the almost Mathieu operator

Several recent works have established dynamical localization for Schrodinger operators, starting From control on the localization length of their eigenfunctions, in terms of their centers of localization. We provide an alternative way to obtain dynamical localization, without resorting to such a strong condition on the exponential decay of the eigenfunctions. Furthermore, we illustrate our purpose with the almost Mathieu operator. H-theta,H-lambda,H-omega = - Delta + lambda cos(2 pi(0 + x omega)), lambda greater than or equal to 15 and omega with good Diophantine properties. More precisely, for almost all theta, for all q > 0, and for all functions psi is an element of l(2)(Z) of compact support, we show that sup(t) [e(-itH0, lambda, omega) psi, \X\(q) e(-ith theta, lambda, omega) psi] < C-psi The proof applies equally well to discrete and continuous random Hamiltonians. In all cases, it uses as input a repulsion principle of singular boxes, supplied in the random case by the multi-scale analysis.