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

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.