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

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 $$\mathrm{i}{\mathrm{\partial }}_{t}u=-{\mathrm{\partial }}_x^{2}u+x^{2}u+\epsilon ⟨x{⟩}^{\mu }\sum _{k\in \mathrm{\Lambda }}(a_k(\omega t)\sin (k\mathrm{\mid }x{\mathrm{\mid }}^{\beta })+b_k(\omega t)\cos (k\mathrm{\mid }x{\mathrm{\mid }}^{\beta }))u,u=u(t,x),x\in \mathbb{R},\beta >1,$$ can be reduced in ${\mathcal{H}}^1(\mathbb{R})$ to an autonomous system for most values of the frequency vector $\omega$, where $\mathrm{\Lambda }\subset \mathbb{R}\setminus \{0\}$, $\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }<\mathrm{\infty }$ and $⟨x⟩:=\sqrt{1+x^2}$. The functions $a_k(\theta )$ and $b_k(\theta )$ are analytic on ${\mathbb{T}}_{\sigma }^n$ and $\mu \ge 0$ 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 $\beta >1$.