论文标题
1-D量子谐波振荡器与时间准周期性二次扰动：Sobolev Norms的降低和生长
1-d Quantum Harmonic Oscillator with Time Quasi-periodic Quadratic Perturbation: Reducibility and Growth of Sobolev Norms
论文作者
论文摘要
For a family of 1-d quantum harmonic oscillator with a perturbation which is $C^2$ parametrized by $E\in{\mathcal I}\subset{\Bbb R}$ and quadratic on $x$ and $-{\rm i}\partial_x$ with coefficients quasi-periodically depending on time $t$, we show the reducibility （即，与时间无关的共轭）对于A.E. $ e $。作为可降低性的应用，我们描述了Sobolev空间中解决方案的行为： - 有限W.R.T. $ t $对于{\ Mathcal i} $中的“大多数” $ e \始终是正确的。 - 对于“通用”时间依赖性扰动，多项式生长和对无限W.R.T.的指数增长$ e $在$ {\ mathcal i} $的“小”部分中发生$ t $。 给出了确实发生Sobolev Norm的生长的具体示例。
For a family of 1-d quantum harmonic oscillator with a perturbation which is $C^2$ parametrized by $E\in{\mathcal I}\subset{\Bbb R}$ and quadratic on $x$ and $-{\rm i}\partial_x$ 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 solution in Sobolev space: -- Boundedness w.r.t. $t$ is always true for "most" $E\in{\mathcal I}$. -- 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.
