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Reducibility of 1-D quantum harmonic oscillator with new unbounded oscillatory perturbations (2311.18384v1)

Published 30 Nov 2023 in math.AP, math-ph, math.DS, and math.MP

Abstract: 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 $${\rm i}\partial_t u = -\partial_x2 u+x2 u+\epsilon \langle x\rangle\mu\sum_{k\in\Lambda}\left(a_k(\omega t)\sin(k|x|\beta)+b_k(\omega t) \cos(k|x|\beta)\right) u,\quad u=u(t,x),~x\in\mathbb{R},~ \beta>1,$$ 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 $\langle x\rangle:=\sqrt{1+x2}$. The functions $a_k(\theta)$ and $b_k(\theta)$ are analytic on $\mathbb Tn_\sigma$ 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 $\beta>1$.

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