Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schr\"odinger operators

TL;DR

This paper introduces a new concept of $eta$-almost periodicity and establishes quantitative spectral and dynamical bounds for such potentials, providing sharp arithmetic criteria for spectral properties of quasiperiodic Schr"odinger operators.

Contribution

It develops a novel notion of $eta$-almost periodicity and applies it to derive precise spectral and dynamical bounds, advancing understanding of quasiperiodic operators.

Findings

Sharp arithmetic criterion for full spectral dimensionality in positive Lyapunov exponent regime

Arithmetic criteria for zero Lyapunov exponent cases

Applications to Sturmian potentials and the critical almost Mathieu operator

Abstract

We introduce a notion of $\beta$-almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded $\beta$-almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schr\"odinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.