论文标题
线性时间依赖性量子谐波振荡器的通用转运器上的$ \ mathbb r $
Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$
论文作者
论文摘要
在本文中，我们考虑线性，依赖时间的量子谐波schrödinger方程$ i \ partial_t u = \ frac {1} {2} {2} {2}（ - \ partial_x^2 + x^2）自我关节和$2π$定期。我们在$ V（t，x，d）$的主要符号上提供足够的条件，以确保存在弱动荡的解决方案，显示出Sobolev Norms的无限时间增长。这些条件在符号的特定空间中是通用的。这表明，通用，古典的伪数，$2π$ - 周期性扰动引起了不稳定的动态。证明是基于[36]的结果，并且基于假数分化的正常形式和局部能量衰减估计值。这些最后被证明是利用穆雷的积极换向者理论。
In this paper we consider the linear, time dependent quantum Harmonic Schrödinger equation $i \partial_t u= \frac{1}{2} ( - \partial_x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb R$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, selfadjoint, and $2π$ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of weakly turbulent solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Frechet space of symbols. This shows that generic, classical pseudodifferential, $2π$-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre's positive commutator theory.
