 00399 Bambusi,D., Graffi,S.
 Time Quasiperiodic unbounded perturbations of Schr\"odinger
operators and KAM methods
(54K, LaTeX)
Oct 9, 00

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Abstract. We eliminate by KAM methods the time dependence in a class of linear
differential equations in
$\ell^2$ subject to an unbounded, quasiperiodic forcing. This entails
the purepoint nature of the Floquet spectrum of the operator $
H_0+\epsilon P(\om t)$ for $\epsilon$ small. Here $H_0$ is the
onedimensional Schr\"odinger operator
$p^2+V$, $V(x)\sim x^{\alpha}, \alpha >2$ for
$x\to\infty$, the time quasiperiodic perturbation $P$ may grow as
$\displaystyle x^{\beta}, \beta <(\alpha2)/{2}$, and the frequency
vector $\omega$ is non resonant. The proof
extends to infinite dimensional spaces the result valid for
quasiperiodically forced linear differential equations and is based on
Kuksin's estimate of solutions of homological equations with non
constant coefficients.
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