Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

TL;DR

This paper develops a quantitative strong almost reducibility theory for Gevrey quasi-periodic cocycles, leading to new spectral and dynamical results for related Schrödinger operators, using a refined perturbative KAM approach.

Contribution

It introduces a quantitative reducibility framework for Gevrey cocycles and applies it to spectral analysis of Schrödinger operators, extending previous results with a refined KAM scheme.

Findings

Spectral gaps decay sub-exponentially with label

Pure point spectrum with localized eigenfunctions for small potentials

Spectrum forms an interval for certain discrete Schrödinger operators

Abstract

We establish a quantitative version of strong almost reducibility result for $\mathrm{sl}(2,\mathbb{R})$ quasi-periodic cocycle close to a constant in Gevrey class. We prove that, for the quasi-periodic Schr\"odinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling, the long range duality operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases and the spectrum is an interval for discrete Schr\"odinger operator acting on $ \mathbb{Z}^d $ with small separable potentials. All these results are based on a refined KAM scheme, and thus are perturbative.