Time Quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods

Details Time Quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods Time Quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods

2000-10-02

arxiv.org/abs/math-ph/0010002v1

Physics

Mathematical Physics

16 pages, Latex2e

Scientific paper

10.1007/s002200100426

We eliminate by KAM methods the time dependence in a class of linear differential equations in $\ell^2$ subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator $ H_0+\epsilon P(\om t)$ for $\epsilon$ small. Here $H_0$ is the one-dimensional Schr\"odinger operator $p^2+V$, $V(x)\sim |x|^{\alpha}, \alpha >2$ for $|x|\to\infty$, the time quasi--periodic perturbation $P$ may grow as $\displaystyle |x|^{\beta}, \beta <(\alpha-2)/{2}$, 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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