标题： 一维量子简谐振子在新的无界振荡扰动下的可约性 显示英文标题

标题： Reducibility of 1-D quantum harmonic oscillator with new unbounded oscillatory perturbations

摘要： 受\cite{LiangLuo2021}中引理1.7的启发，我们证明了一个类似的引理，该引理基于振荡积分和Langer的转折点理论。 从中我们证明，薛定谔方程$${\rm i}\partial_t u = -\partial_x^2 u+x^2 u+\epsilon \langle x\rangle^\mu\sum_{k\in\Lambda}\left(a_k(\omega t)\sin(k|x|^\beta)+b_k(\omega t) \cos(k|x|^\beta)\right) u,\quad u=u(t,x),~x\in\mathbb{R},~ \beta>1,$$可以在$\mathcal{H}^1(\mathbb{R})$中转化为一个自治系统，对于频率向量$\omega$的大多数值，其中$\Lambda\subset\mathbb R\setminus\{0\}$，$|\Lambda|<\infty$和$\langle x\rangle:=\sqrt{1+x^2}$。 函数$a_k(\theta)$和$b_k(\theta)$在$\mathbb T^n_\sigma$和$\mu\geq 0$上是解析的，且将根据$\beta$的值进行选择。 与\cite{LiangLuo2021}相比，新奇之处在于当$\beta>1$时，振荡积分的相位函数更加退化。 显示英文摘要

摘要： Enlightened by Lemma 1.7 in \cite{LiangLuo2021}, we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schr{\"o}dinger equation $${\rm i}\partial_t u = -\partial_x^2 u+x^2 u+\epsilon \langle x\rangle^\mu\sum_{k\in\Lambda}\left(a_k(\omega t)\sin(k|x|^\beta)+b_k(\omega t) \cos(k|x|^\beta)\right) u,\quad u=u(t,x),~x\in\mathbb{R},~ \beta>1,$$ can be reduced in $\mathcal{H}^1(\mathbb{R})$ to an autonomous system for most values of the frequency vector $\omega$, where $\Lambda\subset\mathbb R\setminus\{0\}$, $|\Lambda|<\infty$ and $\langle x\rangle:=\sqrt{1+x^2}$. The functions $a_k(\theta)$ and $b_k(\theta)$ are analytic on $\mathbb T^n_\sigma$ and $\mu\geq 0$ will be chosen according to the value of $\beta$. Comparing with \cite{LiangLuo2021}, the novelty is that the phase functions of oscillatory integral are more degenerate when $\beta>1$.