Quasiballistic Transport for Discrete One-Dimensional Quasiperidic Schrödinger Operators

Abstract

We obtain (up to logarithmic scaling) the power-law lower bound $M_p(T_k)\gtrsim T_k^{(1-\delta )p}$ on a subsequence $T_k\to \infty$, uniformly across $p>0$, for discrete one-dimensional quasiperiodic Schrödinger operators with frequencies satisfying $\beta (\alpha )>\frac{3}{\delta }{\min }_{\sigma }\gamma$. We achieve this by obtaining a quantitative ballistic lower bound for the Abel-averaged time evolution of general periodic Schrödinger operators in terms of the bandwidths. A similar result without uniformity, which assumes $\beta (\alpha )>\frac{C}{\delta }{\min }_{\sigma }\gamma$, was obtained earlier by Jitomirskaya and Zhang, for an implicit constant $C<\infty$.