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}\dotψ=(-Δ+V(x)+εW(ωt,x,-\text{i}\nabla))ψ,\ \ \ \ \ x\in\mathbb{R}^d \end{equation} is considered, where $\omega \in (0,2\pi {)}^n$, $V(x):={\sum }_{j=1}^d{v}_j^2{x}_j^2,v_j\ge v_0>0$, 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 $\ell \ge 2n-1+\beta ,0<\beta <1$. Using techniques developed by Bambusi-Grébert-Maspero-Robert [\emph{Anal. PDE. 11(3):775-799, 2018}] and Rüssmann [\emph{pages 598--624. Lecture Notes in Phys., Vol. 38, 1975}], the paper shows that for any $|\epsilon |\le {\epsilon }_{\star }(n,\ell )$, 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.

36 pages. arXiv admin note: text overlap with arXiv:1702.05274 by other authors

References in corpus (1)

• KAM for the quantum harmonic oscillator