Log-dimensional spectral properties of one-dimensional quasicrystals

TL;DR

This paper investigates the spectral properties of one-dimensional quasicrystals, establishing a criterion for spectral measure continuity and deriving quantum dynamical bounds for Sturmian potentials across various parameters.

Contribution

It introduces a new criterion for spectral measure continuity in log-Hausdorff measures and applies it to Sturmian potentials to prove quantum dynamical bounds.

Findings

Spectral measures are continuous with respect to log-Hausdorff measures.

Logarithmic quantum dynamical lower bounds are established for Sturmian potentials.

Results hold for all coupling constants and almost all rotation numbers.

Abstract

We consider discrete one-dimensional Schr\"odinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to $\log$-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.