Ballistic Transport and Absolute Continuity of One-Frequency Schrödinger Operators

For the solution $u(t)$ to the discrete Schrödinger equation ${\rm i}\frac{d}{dt}u_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(θ+ nα)u_n(t), \quad n\in\Z,$ with $α\in\R\setminus\Q$ and $V\in C^ω(\T,\R)$, we consider the growth rate with $t$ of its diffusion norm $\langle u(t)\rangle_{p}:=\left(\sum_{n\in\Z}(n^{p}+1) |u_n(t)|^2\right)^\frac12$, and the (non-averaged) transport exponents $β_u^{+}(p) := \limsup_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}, \quad β_u^{-}(p):= \liminf_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}.$ We will show that, if the corresponding Schrödinger operator has purely absolutely continuous spectrum, then $β_{u}^{\pm}(p)=1$, provided that $u(0)$ is well localized.

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