Exact local distribution of the absolutely continuous spectral measure

TL;DR

This paper precisely characterizes the local distribution of the spectral measure for one-frequency Schrödinger operators with small potentials, extending known regularity results and analyzing continuity properties at subcritical energies.

Contribution

It provides a detailed description of the local spectral measure distribution for dense small potentials, including subcritical almost Mathieu operators, and studies stratified H"older continuity.

Findings

Spectral measure exhibits optimal 1/2-Hölder continuity within the spectrum.

Characterization of local spectral measure distribution for dense small potentials.

Analysis of stratified Hölder continuity at subcritical energies.

Abstract

It is well-established that the spectral measure for one-frequency Schr\"odinger operators with Diophantine frequencies exhibits optimal $1/2$-H\"older continuity within the absolutely continuous spectrum. This study extends these findings by precisely characterizing the local distribution of the spectral measure for dense small potentials, including a notable result for any subcritical almost Mathieu operators. Additionally, we investigate the stratified H\"older continuity of the spectral measure at subcritical energies.