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

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.