TITLE: Reducibility of Quasi-Periodic Skew-Products and the Spectrum of Schrödinger Operators
AUTHOR: Joaquim Puig i Sadurní
ADVISOR: Carles Simó i Torres
In this thesis we study the reducibility and other
dynamical properties of linear quasi-periodic skew-products, with
special emphasis on those arising from eigenvalue equations of
one-dimensional quasi-periodic Schrödinger operators. To do so we combine dynamical
and spectral methods to give a unified approach and new results,
both from the dynamical and spectral point of view. As an example of this
combination, in this thesis we prove the ``Ten Martini Problem'', posed in
1981 by Kac & Simon.
The first two chapters contain preliminaries and the other novel results.
In the first one we introduce basic concepts such as quasi-periodic skew-products,
quasi-periodic cocycles, reducibility to constant coefficients and
Sacker-Sell spectral theory. The second chapter focuses on quasi-periodic
Schrödinger operators, both continuous and discrete, and their
eigenvalue equations, which we call quasi-periodic Hill's equations
(in the continuous case) and Harper-like equations (in the discrete case).
The third and fourth chapters deal with the structure of the so-called
``resonance tongues'' in quasi-periodic Hill's equations whose potentials
are real analytic, small and with Diophantine frequencies. From the point
of view of Schrödinger operators, this study is equivalent to that of the structure of
spectral gaps in the spectrum. In the third chapter, normal forms are used
to prove the smoothness of tongue boundaries in a constructive way. In the
fourth, real analyticity is proved, for this and other models, using KAM
techniques. As an application we prove the genericity of ``having all gaps
open'' for quasi-periodic Schrödinger operators.
Fifth and sixth chapters deal with discrete quasi-periodic Schrödinger
operators. In chapter V we prove an old conjecture: the ``Ten Martini
Problem''. Using a combination of dynamical and spectral methods we show
that the spectrum of the ``Almost Mathieu operator'' is a Cantor set
for almost all frequencies and noncritical coupling constant. We also give
a partial answer to the ``Strong (or Dry) Ten Martini Problem''. In Chapter
VI we prove a nonperturbative version of Eliasson's result on the
reducibility of Schrödinger cocycles with real analytic potentials.
Finally we include an appendix where the genericity of divergence of
quasi-periodic Birkhoff Normal forms is proved.
The thesis contains an abstract in Catalan, a table of notation and an index.