A Nonperturbative Eliasson's Reducibility Theorem

Abstract

This paper is concerned with discrete, one-dimensional Schr\"odinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schr\"odinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a $G_\delta$-set of energies whose Schr\"odinger cocycle is not reducible to constant coefficients.