Dynamical Localization II with an Application to the Almost Mathieu Operator

Abstract

Several recent works have established dynamical localization for Schrödinger 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 _θ, λ, ω=−Δ+λ cos(2π(θ+xω)), λ≥15 and ω with good Diophantine properties. More precisely, for almost all θ, for all q>0, and for all functions ψ∈ℓ²($\mathbb{Z}$) of compact support, we show that$\mathop {\sup }\limits_t \left\langle {e^{ - itH_{\theta ,\lambda ,\omega } } \psi ,\left| X \right|^q e^{ - itH_{\theta ,\lambda ,\omega } } \psi } \right\rangle < C\psi$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.