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-Grebert-Maspero-Robert [\emph{Anal. PDE. 11(3):775-799, 2018}] and Russmann [\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.

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具有有限可微扰动的 d 维量子谐振子的可约化性

在本文中，具有伪微分时间准周期扰动的$d$维量子谐振子 \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} 被考虑，其中 $\omega\in(0,2\pi)^n$、$V(x):=\sum_{j=1}^d v_j^2x_j^2、v_j\geq v_0>0$ 和 $ W(\theta,x,\xi)$是$(x,\xi)$中最多为2次的实多项式，其系数属于$\theta\in\mathbb中的$C^{\ell}$ {T}^n$ 满足 $\ell\geq 2n-1+\beta,\0<\beta<1$ 的订单 $\ell$。使用 Bambusi-Grebert-Maspero-Robert [\emph{Anal. 偏微分方程。11(3):775-799, 2018}] 和 Russmann [\emph{第 598--624 页。物理讲义，卷。38, 1975}]，论文表明对于任何 $|\epsilon|\leq \epsilon_{\star}(n,\ell)$，