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 $i \partial_t u =-\partial_x^2 u+x^2 u+ε\langle x\rangle^μ W(νx,ωt)u,\quad u=u(t,x),~x\in\mathbb R,~ 0\leq μ<\frac13,$ can be reduced in $\mathcal H^1(\mathbb R)$ to an autonomous system for most values of the frequency vector $ω$ and $ν$, where $W(φ, θ)$ is a smooth map from $\mathbb T^d\times \mathbb T^n$ to $\mathbb R$ and odd in $φ$.

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