Reducibility or non-uniform hyperbolicity for quasiperiodic Schrodinger cocycles

TL;DR

This paper proves that for most frequencies and energies, quasiperiodic Schrödinger cocycles are either reducible or nonuniformly hyperbolic, providing insights into the spectrum's nature and confirming the Aubry-Andre conjecture.

Contribution

It establishes a dichotomy for quasiperiodic Schrödinger cocycles for almost all frequencies and energies, advancing understanding of spectral properties and completing the Aubry-Andre conjecture proof.

Findings

Almost every frequency and energy leads to reducible or nonuniformly hyperbolic cocycles.

The results give control over the absolutely continuous spectrum.

The proof completes the Aubry-Andre conjecture on the spectrum measure.

Abstract

We show that for almost every frequency alpha \in \R \setminus \Q, for every C^omega potential v:\R/\Z \to R, and for almost every energy E the corresponding quasiperiodic Schrodinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrodinger operator, and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator.