论文标题
Ehrenfest时间尺度上的Van Vleck公式和用于频率相的固定相渐近
The Van Vleck Formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phases
论文作者
论文摘要
Van Vleck公式是与与时间依赖性Schrödinger方程相关的传播器的积分内核的半经典近似。在适当的假设下，我们对此近似值进行了严格的处理，该处理在“ Ehrenfest时间尺度”上有效，即$ \ hbar $依赖性时间间隔，最常见的是以$ | t |的形式使用。 \ leq c | \ log \ hbar | $。我们的派生基于对积分内核的近似，通常称为“ Herman-Kluk近似”，该近似值将内核视为相位空间中点参数的高斯参数的高斯的积分叠加。正如罗伯特（Robert）所表明的那样，这在Ehrenfest时间间隔上产生了有效的近似值。为了从Herman-Kluk近似中得出Van Vleck的近似，我们被引导开发固定相的渐进型，其中相位函数以非平凡的方式取决于频率参数，这一结果可能具有独立的兴趣。
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.
