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arXiv · 2003.12951

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

Zhenguo Liang·Jiawen Luo
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Abstract

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 $\varphi$.

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Zhenguo Liang, Jiawen Luo. 2020-03-29. Reducibility of 1-d Quantum Harmonic Oscillator Equation with Unbounded Oscillation Perturbations. https://arxiv.org/abs/2003.12951

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