Ballistic transport in one-dimensional quasi-periodic continuous Schrödinger equation

For the solution q(t) to the one-dimensional continuous Schrödinger equationi∂tq(x,t)=−∂x2q(x,t)+V(ωx)q(x,t),x∈R, with ω∈Rd satisfying a Diophantine condition, and V a real-analytic function on Td, we consider the growth rate of the diffusion norm ‖q(t)‖D:=(∫Rx2|q(x,t)|2dx)12 for any non-zero initial condition q(0)∈H1(R) with ‖q(0)‖D<∞. We prove that ‖q(t)‖D grows linearly with t if V is sufficiently small.