In this article, the second of three, we discuss and develop the basis of a Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type models. This Weyl quantisation may serve as the main mathematical tool to implement the program of space adiabatic perturbation theory in such models. As we already argued in our first article, space adiabatic perturbation theory offers an ideal framework to overcome the obstacles that hinder the direct implementation of the conventional Born-Oppenheimer approach in the canonical formulation of loop quantum gravity. Additionally, we conjecture the existence of a new form of the Segal-Bargmann-Hall "coherent state" transform for compact Lie groups $G$, which we prove for $G=U(1)^{n}$ and support by numerical evidence for $G=SU(2)$. The reason for conjoining this conjecture with the main topic of this article originates in the observation, that the coherent state transform can be used as a basic building block of a coherent state quantisation (Berezin quantisation) for compact Lie groups $G$. But, as Weyl and Berezin quantisation for $\mathbb{R}^{2d}$ are intimately related by heat kernel evolution, it is natural to ask, whether a similar connection exists for compact Lie groups, as well. Moreover, since the formulation of space adiabatic perturbation theory requires a (deformation) quantisation as minimal input, we analyse the question to what extent the coherent state quantisation, defined by the Segal-Bargmann-Hall transform, can serve as basis of the former.
A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates
本文构造了分子量子力学的指数精度含时Born-Oppenheimer近似。我们研究的分子系统的电子质量是固定的，其核质量是成正比的$ε^{-4}$，其中$ε$是一个小的扩展参数。通过对渐近展开式的最佳截断，我们构造了含时薛定谔方程的近似解，对于某些C和$γ>0$，该近似解与精确的归一化解一致，直到其范数由$\ds C \exp(-γ/ε^2)$限制的误差为止.
Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area
本文研究了当势函数V在无穷远处不趋于无穷大时，薛定谔算子本征值的渐近性。这种势称为简并势。关键是相空间中的一组相关联的哈密顿量小于一个固定的能量E可能有一个无限大的体积，使外尔公式给出的行为的计数功能必须重新审视。我们回顾在这方面的各种结果，在经典的背景下，以及在半经典的和评论不同的方法。在第3，4节中，我们提出了我们与A Morame（Université de Nantes）的联合工作，涉及退化势V（x）=f（y）g（z），其中g被假定为m个变量的齐次正函数，f是n个变量的光滑严格正函数，最小值在0。当f在无穷远处趋于无穷大时，我们给出了小于固定能量的特征值个数的半经典渐近性质。然后，我们给出了一个尖锐的估计的低本征值，使用玻恩奥本海默近似。通过一种改进的方法，我们也定位了更高的能量。最后，我们将前面的方法应用到一类在正则超曲面上为零的势函数上。
Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；University of Oslo（奥斯陆大学）；Kungl. Tekniska Högskolan（皇家理工学院）；University of Delaware（特拉华大学）
