Title: Resonance tongues in the Quasi-Periodic Hill-Schr\"odinger Equation with three frequencies

Authors: Joaquim Puig (1) and Carles Sim\'o (2)
(1)Departament de Matem\`atica Aplicada I. Universitat Polit\`ecnica de Catalunya. 
Dia\-gonal, 647. 08028 Barcelona (Spain).
(2) Departament de Matem\`atica Aplicada i An\`alisi. Universitat de Barcelona. 
Gran Via de les Corts Catalanes, 585. 08007 Barcelona (Spain).

Date: July 2, 2010. To appear in Regul. Chaotic Dyn.


Abstract:
In this paper we investigate numerically  the following Hill's equation
$x''+(a+bq(t))x=0$ where $q(t)=\cos{t}+\cos{\sqrt{2}t}+\cos{\sqrt{3}t}$ is a
quasi-periodic forcing with three rationally independent frequencies. It
appears,also, as the eigenvalue equation of a Schr\"odinger operator with
quasi-periodic potential.

Massive numerical computations were performed for the rotation number and the
Lyapunov exponent in order to detect open and collapsed gaps, resonance
tongues. Our results show that the quasi-periodic case with three independent
frequencies is very different not only from the periodic analogs, but also
from the case of two frequencies. Indeed, for large values of $b$ the
spectrum contains open intervals at the bottom. From a dynamical point of
view we numerically give evidence of the existence of open intervals of $a$,
for large $b$ where the system is nonuniformly hyperbolic: the system does
not have an exponential dichotomy but the Lyapunov exponent is positive. In
contrast with the region with zero Lyapunov exponents, both the rotation
number and the Lyapunov exponent do not seem to have square root behavior at
endpoints of gaps. The rate of convergence to the rotation number and the
Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be
different from the reducible case.

Keywords: 
Quasi-Periodic Schr\"odinger operators, Quasi-Periodic Cocycles and
Skew-products, Spectral Gaps, Resonance Tongues, Rotation number, Lyapunov
exponent, Numerical explorations}