$\mathbb R$上线性不定常量子谐振子的一般输运子
Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$
摘要：
本文考虑线性含不定常量子调和薛定谔方程$i \partial_t u= \frac{1}{2} ( - \partial_x^2 + x^2) u + V(t, x, D)u$，$x \in \mathbb R$，其中$V(t,x,D)$是0阶自伴的经典伪微分算子，$2π$在时间上是周期的.我们给出了$V(t,x,D)$的主象征上的充分条件，保证了Sobolev范数无限增长的弱湍流解的存在性。这些条件在符号的弗雷歇空间中是通用的。这表明，通用的，经典的伪微分，$2π$周期扰动引起不稳定的动态。证明建立在[36]的结果上，并且它基于伪微分范式和局部能量衰减估计。这些最后证明了利用Zerdre的积极换向器理论。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
