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.