TITLE: Analytic families of reducible linear quasi-periodic differential equations
AUTHORS: Joaquim Puig \& Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi,
Univ. de Barcelona, Gran Via 585, 08007 Barcelona, Spain
puig@maia.ub.es, carles@maia.ub.es
ABSTRACT:
In this paper we study the existence of analytic families of
reducible linear quasi-periodic differential equations in
matrix Lie algebras. Under suitable conditions we show, by
means of a KAM scheme, that a real analytic quasi-periodic
system close to a constant matrix can be modified by the
addition of a time-free matrix which makes it reducible
to constant coefficients. If the system depends analytically on
external parameters, then this modifying term is also analytic.
As a major application, we prove the analyticity of resonance
tongue boundaries in Hill's equation with a small quasi-periodic
forcing. Several consequences for the spectrum of Schr\"odinger
operators with quasi-periodic forcing are derived. In
particular, we prove that, generically, the spectrum of
Schr\"odinger operators with a small real analytic and
quasi-periodic potential has all spectral gaps open and,
therefore, it is a Cantor set. Some other applications are
included for linear quasi-periodic systems on $so(3,\mR)$ and
$sp(n,\mR)$.