Ballistic Transport in One-Dimensional Quasi-Periodic Continuous Schr\"odinger Equation

TL;DR

This paper proves that solutions to a one-dimensional quasi-periodic Schrödinger equation exhibit linear growth in their diffusion norm over time when the potential is sufficiently small, indicating ballistic transport behavior.

Contribution

It establishes the linear growth of the diffusion norm for the continuous Schrödinger equation with quasi-periodic potential under smallness conditions, demonstrating ballistic transport in this setting.

Findings

Diffusion norm grows linearly with time for small potentials.

Solutions exhibit ballistic transport behavior.

Results apply to Diophantine frequency conditions.

Abstract

For the solution $q(t)$ to the one-dimensional continuous Schr\"odinger equation $${\rm i}\partial_t{q}(x,t)=-\partial_x^2 q(x,t) + V(\omega x) q(x,t), \quad x\in{\Bbb R},$$ with $\omega\in{\Bbb R}^d$ satisfying a Diophantine condition, and $V$ a real-analytic function on ${\Bbb T}^d$, we consider the growth rate of the diffusion norm $\|q(t)\|_{D}:=\left(\int_{\Bbb R}x^2|q(x,t)|^2dx\right)^{\frac12}$ for any non-zero initial condition $q(0)\in H^{1}({\Bbb R})$ with $\|q(0)\|_D<\infty$. We prove that $\|q(t)\|_{D}$ grows {\it linearly} with $t$ if $V$ is sufficiently small.