论文标题
Sobolev规范的几乎可降低和振荡增长
Almost reducibility and oscillatory growth of Sobolev norms
论文作者
论文摘要
对于1D量子谐波振荡器，由$（x， - {\ rm i} \ partial_x）$的时间的Quasi periodic二次形式扰动，我们显示其几乎可降低。基于几乎降低的方案，描述了溶液规范的生长。特别是，如果方程不可还原，则显示$ o（t^s） - $上限为$ \ ch^s- $ norm。此外，通过Anosov-katok的结构，我们还展示了这种上限的最佳性，即存在$ {\ Mathcal H}^s- $ norm $ o（t^s）$ o（t^s）$的$ {\ Mathcal H}^s- $ norm的成长为$ t \ toftty $ to \ inftty $ to $ to $ to $ to $ t^$ t^$ t^$ t^$ t^$ t^s^ost^s^os的$ o（t^s）的增长。
For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-{\rm i}\partial_x)$, we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an $o(t^s)-$upper bound is shown for the $\CH^s-$norm if the equation is non-reducible. Moreover, by Anosov-Katok construction, we also show the optimality of this upper bound, i.e., the existence of quasi-periodic quadratic perturbation for which the growth of ${\mathcal H}^s-$norm of the solution is $o(t^s)$ as $t\to\infty$ but arbitrarily ``close" to $t^s$ in an oscillatory way.
