Ballistic Transport and Absolute Continuity of One-Frequency Schr\"{o}dinger Operators

TL;DR

This paper demonstrates that for one-frequency Schrödinger operators with purely absolutely continuous spectrum, solutions exhibit ballistic transport characterized by transport exponents equal to 1, assuming initial localization.

Contribution

It establishes a direct link between purely absolutely continuous spectrum and ballistic transport in discrete Schrödinger equations with quasi-periodic potentials.

Findings

Transport exponents equal to 1 for localized initial data

Purely absolutely continuous spectrum implies ballistic transport

Results apply to one-frequency quasi-periodic Schrödinger operators

Abstract

For the solution $u(t)$ to the discrete Schr\"odinger equation $${\rm i}\frac{d}{dt}u_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(\theta + n\alpha)u_n(t), \quad n\in\Z,$$ with $\alpha\in\R\setminus\Q$ and $V\in C^\omega(\T,\R)$, 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 $$\beta_u^{+}(p) := \limsup_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}, \quad \beta_u^{-}(p):= \liminf_{t \to \infty} \frac{2\log \langle u(t)\rangle_{p}}{p\log t}.$$ We will show that, if the corresponding Schr\"odinger operator has purely absolutely continuous spectrum, then $\beta_{u}^{\pm}(p)=1$, provided that $u(0)$ is well localized.