Birkhoff normal form in low regularity for the nonlinear quantum harmonic oscillator

TL;DR

This paper demonstrates that for the nonlinear quantum harmonic oscillator with generic potentials, low-energy modes remain nearly unchanged over long times, using Birkhoff normal form techniques in low regularity settings.

Contribution

It extends Birkhoff normal form methods to low regularity for the nonlinear quantum harmonic oscillator with generic potentials, showing long-time stability of low modes.

Findings

Low modes are almost preserved over very long times.

Results hold for almost all potentials V.

Applicable in low regularity Sobolev spaces.

Abstract

Given small initial solutions of the nonlinear quantum harmonic oscillator on $\mathbb{R}$, we are interested in their long time behavior in the energy space which is an adapted Sobolev space. We perturbate the linear part by $V$ taken as multiplicative potentials, in a way that the linear frequencies satisfy a non-resonance condition. More precisely, we prove that for almost all potentials $V$, the low modes of the solution are almost preserved for very long times.