Reducibility of 1-d Quantum Harmonic Oscillator Equation with Unbounded Oscillation Perturbations

We build a new estimate relative with Hermite functions based upon oscillatory integrals and Langer's turning point theory. From it we show that the equation $$i \partial_t u =-\partial_x^2 u+x^2 u+\epsilon \langle x\rangle^{\mu} W(\nu x,\omega t)u,\quad u=u(t,x),~x\in\mathbb R,~ 0\leq \mu<\frac13,$$ can be reduced in $\mathcal H^1(\mathbb R)$ to an autonomous system for most values of the frequency vector $\omega$ and $\nu$, where $W(\varphi, \theta)$ is a smooth map from $\mathbb T^d\times \mathbb T^n$ to $\mathbb R$ and odd in $\varphi$.