Linear Schr\"odinger equation with an almost periodic potential

TL;DR

This paper proves that a linear Schr"odinger equation with a small, almost-periodic perturbation can be transformed into a simpler form, enabling long-term control of solutions' norms.

Contribution

It demonstrates the reducibility of the Schr"odinger equation with almost-periodic perturbations under specific conditions, extending previous results to unbounded and analytic cases.

Findings

Equation is reducible to constant coefficients

Long-term control of Sobolev and analytic norms

Applicable to small, unbounded, almost-periodic perturbations

Abstract

We study the reducibility of a Linear Schr\"odinger equation subject to a small unbounded almost-periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity and on the frequency of the almost-periodic perturbation, we prove that such an equation is reducible to constant coefficients via an analytic almost-periodic change of variables. This implies control of both Sobolev and Analytic norms for the solution of the corresponding Schr\"odinger equation for all times.