For a family of 1-d quantum harmonic oscillators with a perturbation which is $C^2$ parametrized by $E\in \mathcal{I}\subset \mathbb{R}$ and quadratic on x and $-\mathrm{i}{\partial }_x$ with coefficients quasi-periodically depending on time t, we show the reducibility (i.e., conjugation to time-independent) for a.e. E. As an application of reducibility, we describe the behaviors of solutions in Sobolev space:

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Boundedness w.r.t. t is always true for “most” $E\in \mathcal{I}$.

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For “generic” time-dependent perturbation, polynomial growth and exponential growth to infinity w.r.t. t occur for E in a “small” part of $\mathcal{I}$.

Concrete examples are given for which the growths of Sobolev norm do occur.

中文翻译：

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具有时间准周期二次扰动的一维量子谐波振荡器：Sobolev范数的可约性和增长

对于一类具有扰动的一维量子谐波振荡器 $C^2$ 被参数化 $Ë\in \mathcal{一世}\subset \mathbb{[R}$并在x和$-\mathrm{一世}{\partial }_X$利用准周期取决于时间t的系数，我们显示了ae E的可约化性（即与时间无关的共轭）。作为可简化性的一种应用，我们描述了Sobolev空间中解的行为：

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有界WRT牛逼总是真实的“最”$Ë\in \mathcal{一世}$。

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为“通用”时微扰，多项式生长和指数增长到无穷大WRT吨发生为Ë中的“小”部分$\mathcal{一世}$。

给出了发生Sobolev范数增长的具体例子。