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 $$i{\mathrm{\partial }}_{t}u=-{\mathrm{\partial }}_x^{2}u+x^{2}u+ϵ⟨x{⟩}^{\mu }W(\nu x,\omega t)u,u=u(t,x),x\in \mathbb{R},0\le \mu <\frac{1}{3},$$ can be reduced in ${\mathcal{H}}^1(\mathbb{R})$ to an autonomous system for most values of the frequency vector $\omega$ and $\nu$, where $W(\phi ,\theta )$ is a smooth map from ${\mathbb{T}}^d\times {\mathbb{T}}^n$ to $\mathbb{R}$ and odd in $\phi$.