关于Herman-Kluk半经典逼近
On the Herman-Kluk Semiclassical Approximation
摘要：
对于$\R^d\times\R^d = T^*(\R^d)$上的次二次符号$H$，当$\hat H=H(x,\hbar D_x)$（$H$的$\hbar$ -Weyl量子化）时，含时薛定谔方程$i\hbar\frac{\partialψ}{\partial t} = \hat Hψ$的量子传播子是半经典傅里叶积分算子。其Schwartz核由二次相位和振幅描述。在每一个时间$t$，当$\hbar$是小的，它是“基本上支持”在由$H$生成的经典流的图形的邻域中，具有在$\hbar$中的振幅的完全一致的渐近展开。在本文中，我们的目标是重新审视这一众所周知的和fundamental的结果，强调的灵活性，选择一个二次复相位函数和全球$L^2$估计时，$\hbar$是小的，时间$t$是大的。在化学物理学中，最简单的相位选择被称为赫尔曼-克鲁克公式。证明了传播子的半经典展开式对$| t| << \frac{1}{4δ}|\log\hbar|$是有效的，其中$δ>0$是经典系统的稳定性参数.
英文摘要：
For a subquadratic symbol $H$ on $\R^d\times\R^d = T^*(\R^d)$, the quantum propagator of the time dependent Schrödinger equation $i\hbar\frac{\partialψ}{\partial t} = \hat Hψ$ is a Semiclassical Fourier-Integral Operator when $\hat H=H(x,\hbar D_x)$ ($\hbar$-Weyl quantization of $H$). Its Schwartz kernel is describe by a quadratic phase and an amplitude. At every time $t$, when $\hbar$ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by $H$, with a full uniform asymptotic expansion in $\hbar$ for the amplitude. In this paper our goal is to revisit this well known and fondamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global $L^2$ estimates when $\hbar$ is small and time $t$ is large. One of the simplest choice of the phase is known in chemical physics as Herman-Kluk formula. Moreover we prove that the semiclassical expansion for the propagator is valid for $| t| << \frac{1}{4δ}|\log\hbar|$ where $δ>0$ is a stability parameter for the classical system.
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