Time Quasi-periodic unbounded perturbations of Schr\"odinger operators and KAM methods

TL;DR

This paper uses KAM methods to remove time dependence in certain unbounded, quasi-periodic Schrödinger equations, proving the pure-point spectrum of the perturbed operator for small perturbations.

Contribution

It extends KAM techniques to infinite-dimensional Schrödinger operators with unbounded, quasi-periodic forcing, establishing spectral properties under new conditions.

Findings

Proves pure-point spectrum for small perturbations of unbounded Schrödinger operators.

Extends KAM methods to infinite-dimensional, unbounded operator settings.

Demonstrates the elimination of time dependence in a class of linear differential equations.

Abstract

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.