Escherichfest时间尺度上的Van Vleck公式和频率相关相位的稳相渐近性
The Van Vleck Formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phases
摘要：
Van Vleck公式是与时间相关的薛定谔方程相关的传播子的积分核的半经典近似。在适当的假设下，我们提出了一个严格的治疗这种近似是有效的“Escherfest时间尺度“，即$\hbar$依赖的时间间隔，最常见的形式$|t| \leq c|\log\hbar|$。我们的推导是基于一个近似的积分内核通常被称为“赫尔曼-克鲁克近似”，实现内核作为一个整体的叠加高斯参数化的点在相空间。正如罗伯特所示，这产生了在Ehrfest时间间隔上的有效近似。为了从Herman-Kluk近似导出Van Vleck近似，我们导出了定相渐近，其中相位函数以非平凡的方式依赖于频率参数，这一结果可能是独立的。
英文摘要：
The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on "Ehrenfest time scales", i.e. $\hbar$-dependent time intervals which most commonly take the form $|t| \leq c|\log\hbar|$. Our derivation is based on an approximation to the integral kernel often called the "Herman-Kluk approximation", which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert, this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman-Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.
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