Quantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropy

Abstract

In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schrödinger operators $H_{f,\hat{I}¸}u(n)=u(n+1)+u(n-1)+\ddot{I}•(f^n\hat{I}¸)u(n)$, where $\ddot{I}•:\mathcal{M}\to \mathbb{R}$ is a piecewise Hölder function on a compact Riemannian manifold $\mathcal{M}$, and $f:\mathcal{M}\to \mathcal{M}$ is a uniquely ergodic volume preserving map with zero topological entropy. As corollaries we obtain localization-type statements for shifts and skew-shifts on higher dimensional tori with arithmetic conditions on the parameters. These are the first localization-type results with precise arithmetic conditions for multi-frequency quasiperiodic and skew-shift potentials.

30 pages