Reducibility of the quantum harmonic oscillator in $d$-dimensions with finitely differentiable perturbations

TL;DR

This paper proves that a $d$-dimensional quantum harmonic oscillator with finitely differentiable quasi-periodic perturbations can be transformed into a simpler form for most frequencies, extending reducibility results to less smooth perturbations.

Contribution

It extends reducibility results of quantum harmonic oscillators to finitely differentiable perturbations using advanced KAM techniques.

Findings

Most frequencies lead to reducibility of the system.

The set of non-reducible frequencies has small measure.

The results apply to perturbations with limited smoothness.

Abstract

In this paper, the $d$-dimensional quantum harmonic oscillator with a pseudo-differential time quasi-periodic perturbation \begin{equation}\label{0} \text{i}\dot{\psi}=(-\Delta+V(x)+\epsilon W(\omega t,x,-\text{i}\nabla))\psi,\ \ \ \ \ x\in\mathbb{R}^d \end{equation} is considered, where $\omega\in(0,2\pi)^n$, $V(x):=\sum_{j=1}^d v_j^2x_j^2, v_j\geq v_0>0$, and $W(\theta,x,\xi)$ is a real polynomial in $(x,\xi)$ of degree at most two, with coefficients belonging to $C^{\ell}$ in $\theta\in\mathbb{T}^n$ for the order $\ell$ satisfying $\ell\geq 2n-1+\beta,\ 0<\beta<1$. Using techniques developed by Bambusi-Gr\'ebert-Maspero-Robert [\emph{{Anal. PDE. 11(3):775-799, 2018}}] and R\"ussmann [\emph{pages 598--624. Lecture Notes in Phys., Vol. 38, 1975}], the paper shows that for any $|\epsilon|\leq \epsilon_{\star}(n,\ell)$, there is a set $\mathcal{D}_{\epsilon}\subset (0,2\pi)^n$ with big Lebesgue measure, such that for any $\omega \in\mathcal{D}_{\epsilon}$, the system is reducible.