Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase

TL;DR

This paper establishes a universal hierarchical structure in the eigenfunctions of quasiperiodic operators, revealing a sharp spectral transition and detailed asymptotics, advancing understanding of localization phenomena.

Contribution

It introduces a new hierarchical framework governed by exponential phase resonances, providing exact asymptotics and characterizing the spectral transition in phase for almost Mathieu operators.

Findings

Sharp spectral transition between localization and singular continuous spectrum.

Universal hierarchical structure of eigenfunctions governed by exponential phase resonances.

Exact exponential asymptotics of eigenfunctions and transfer matrices.

Abstract

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.