Logarithmic quantum dynamical bounds for arithmetically defined ergodic Schr\"odinger operators with smooth potentials

TL;DR

This paper introduces a method to derive power-logarithmic bounds on the growth of moments of the position operator in one-dimensional ergodic Schrödinger operators with smooth potentials, using Bourgain's semi-algebraic approach.

Contribution

It provides a novel technique to establish logarithmic bounds for quantum dynamical systems with complex underlying dynamics and arithmetic conditions.

Findings

Established power-logarithmic bounds for moments growth

Applied Bourgain's semi-algebraic method to ergodic Schrödinger operators

Extended results to operators with multifrequency shift or skew shift dynamics

Abstract

We present a method for obtaining power-logarithmic bounds on the growth of the moments of the position operator for one-dimensional ergodic Schr\"odinger operators. We use Bourgain's semi-algebraic method to obtain such bounds for operators with multifrequency shift or skew shift underlying dynamics with arithmetic conditions on the parameters.