Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger 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.