Resonance tongues and instability pockets in the quasi--periodic
Hill--Schr\"odinger equation

Henk Broer(1), Joaquim Puig(2) and Carles Sim\'o(2)

(1) Dept. of Mathematics and Comp. Sci., Univ. of Groningen, PO Box 800, 
9700 AV Groningen, The Netherlands

(2) Dept. de Matem\`atica Aplicada i An\`alisi, Univ. de Barcelona, 
Gran Via 585, 08007 Barcelona, Spain

Abstract

This paper concerns Hill's equation with a (parametric) forcing that
is real analytic, quasi-periodic with frequency vector $\om,$ and
even in the time $t.$ Schr\"odinger's equation with quasi-periodic potential
is a particular case. We include small parameters $b_1,\ldots,b_p$ 
and a `frequency' parameter $a.$
In the parameter space $\mR^{p+1} = \{a,b\},$ for small values of $|b|$
the following holds. The `tongues' with rotation number
$\halfje \langle \bk,\om \rangle,$ $\bk \in \mZ^d,$
have $C^\infty$-boundaries.
Our arguments are based on reducibility, averaging and 
certain spectral properties of the Schr\"odinger
operator with quasi-periodic potential.
Analogous to the periodic case, further results are obtained, 
such as a criterion for transversality of the tongue boundaries at $b=0$, 
the occurrence of instability pockets in $\mR^{p+1} = \{a,b\},$
corresponding with spectral gaps for the Schr\"odinger operator and 
results on the behaviour of Lyapunov exponent and rotation number for
fixed $b.$