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

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.