Joaquim Puig
A Nonperturbative Eliasson's Reducibility Theorem
(345K, pdf)

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.