Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schr\"odinger Equation

TL;DR

This paper proves that solutions to a one-dimensional quasi-periodic discrete Schrödinger equation exhibit ballistic transport, with the diffusion norm growing linearly over time when the potential is sufficiently small.

Contribution

It establishes linear growth of the diffusion norm for all initial phases in the small potential regime, demonstrating ballistic motion in a quasi-periodic setting.

Findings

Diffusion norm grows linearly with time.

Ballistic transport occurs for all phases under small potential.

Results hold for Diophantine frequencies and real-analytic potentials.

Abstract

For the solution $q(t)=(q_n(t))_{n\in\mathbb Z}$ to one-dimensional discrete Schr\"odinger equation $${\rm i}\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n, \quad n\in\mathbb Z,$$ with $\omega\in\mathbb R^d$ Diophantine, and $V$ a small real-analytic function on $\mathbb T^d$, we consider the growth rate of the diffusion norm $\|q(t)\|_{D}:=\left(\sum_{n}n^2|q_n(t)|^2\right)^{\frac12}$ for any non-zero $q(0)$ with $\|q(0)\|_{D}<\infty$. We prove that $\|q(t)\|_{D}$ grows {\it linearly} with the time $t$ for any $\theta\in\mathbb T^d$ if $V$ is sufficiently small.