Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

Abstract

We establish a quantitative version of strong almost reducibility result for sl(2, R) quasi-periodic cocycle close to a constant in Gevrey class. We prove that, for the quasi-periodic Schrödinger operators with small Gevrey potentials, the length of spectral gaps decays subexponentially 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 Schrödinger operator acting on Z d with small separable potentials. All these results are based on a refined KAM scheme, and thus are perturbative.