Non-perturbative localization on the strip and Avila's almost reducibility conjecture

TL;DR

This paper proves non-perturbative localization for quasi-periodic operators and confirms Avila's almost reducibility conjecture for Schrödinger operators with Diophantine frequencies, advancing understanding in spectral theory.

Contribution

It establishes non-perturbative localization results and proves Avila's almost reducibility conjecture for a broad class of Schrödinger operators with trigonometric potentials.

Findings

Proved non-perturbative Anderson localization on the strip.

Established Avila's almost reducibility conjecture for all Diophantine frequencies.

Derived a non-selfadjoint version of Haro and Puig's formula.

Abstract

We prove non-perturbative Anderson localization and almost localization for a family of quasi-periodic operators on the strip. As an application we establish Avila's almost reducibility conjecture for Schr\"odinger operators with trigonometric potentials and all Diophantine frequencies, whose proof for analytic potentials was announced in Avila's 2015 Acta paper. As part of our analysis, we derive a non-selfadjoint version of Haro and Puig's formula connecting Lyapunov exponents of the dual model to those of the original operator.