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

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{ö}dinger equation $${\rm i}\partial_t u = -\partial_x^2 u+x^2 u+ε\langle x\rangle^μ\sum_{k\inΛ}\left(a_k(ωt)\sin(k|x|^β)+b_k(ωt) \cos(k|x|^β)\right) u,\quad u=u(t,x),~x\in\mathbb{R},~ β>1,$$ can be reduced in $\mathcal{H}^1(\mathbb{R})$ to an autonomous system for most values of the frequency vector $ω$, where $Λ\subset\mathbb R\setminus\{0\}$, $|Λ|<\infty$ and $\langle x\rangle:=\sqrt{1+x^2}$. The functions $a_k(θ)$ and $b_k(θ)$ are analytic on $\mathbb T^n_σ$ and $μ\geq 0$ will be chosen according to the value of $β$. Comparing with \cite{LiangLuo2021}, the novelty is that the phase functions of oscillatory integral are more degenerate when $β>1$.