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

TL;DR

This paper establishes upper quantum dynamical bounds for Schrödinger operators with potentials generated by uniquely ergodic maps of zero topological entropy, leading to new localization results under specific arithmetic conditions.

Contribution

It provides the first localization-type results with precise arithmetic conditions for multi-frequency quasiperiodic and skew-shift potentials.

Findings

Derived upper quantum dynamical bounds from positive Lyapunov exponents.

Achieved localization-type results for higher-dimensional tori shifts and skew-shifts.

Established new results for potentials with underlying zero topological entropy dynamics.

Abstract

In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schr\"odinger operators $H_{f,\theta} u(n)=u(n+1)+u(n-1)+ \phi(f^n\theta)u(n)$, where $\phi : \mathcal{M}\to {\Bbb R}$ is a piecewise H\"older 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.