Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrodinger Equation

For the solution $q(t)=(q_n(t))_{ninmathbb Z}$ to one-dimensional discrete Schrodinger equation $${rm i}dot{q}_n=-(q_{n+1}+q_{n-1})+ V(theta+nomega) q_n, quad ninmathbb Z,$$ with $omegainmathbb R^d$ Diophantine, and $V$ a small real-analytic function on $mathbb T^d$, we consider the growth rate of the diffusion norm $|q(t)|_{D}:=left(sum_{n}n^2|q_n(t)|^2right)^{frac12}$ 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 $thetainmathbb T^d$ if $V$ is sufficiently small.