Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators

TL;DR

This paper proves sharp Holder continuity results for the spectral measure of one-frequency Schrödinger operators with Diophantine frequencies, especially near almost reducible energies and for certain potentials.

Contribution

It provides new sharp continuity bounds for spectral measures, extending results to all energies for specific cases like the almost Mathieu operator with subcritical coupling.

Findings

1/2-Holder continuity near almost reducible energies

Results apply to all energies for small potentials in certain models

Sharp bounds on the modulus of continuity of spectral measures

Abstract

We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrodinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Holder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.