Quasiballistic transport for discrete one-dimensional quasiperiodic Schrödinger operators

Abstract

For discrete one-dimensional quasiperiodic Schrödinger operators with frequencies satisfying $\beta (\alpha )>\left(\frac{3}{\delta }\right){\min }_{\sigma }\gamma$, we obtain (up to logarithmic scaling) the power-law lower bound $M_p(T_k)\gtrsim T_k^{(1-\delta )p}$ on a subsequence $T_k\to \infty$, where $\gamma$ is the associated Lyapunov exponent and $\sigma$ is the spectrum. We achieve this by obtaining a quantitative ballistic lower bound for the Abel-averaged entries of the time evolution operator associated with general periodic Schrödinger operators in terms of the bandwidths. A similar result which assumes $\beta (\alpha )>\left(\frac{C}{\delta }\right){\min }_{\sigma }\gamma$, was obtained earlier by Jitomirskaya and Zhang, for an implicit constant $C<\infty$.