相空间反射算符驱动的能量跃迁
Energy transitions driven by phase space reflection operators
摘要：
相空间反射算子是密度算子和观测量的Wigner-Weyl表示的核心。相应的经典反射的作用是已知的在构造半经典近似的纯本征态和它们的粗粒度微正则叠加的维格纳函数，这不限于经典的可积系统。在它们作为幺正算符的积极作用中，它们产生由跃迁维格纳函数（或交叉维格纳函数）指定的本征态对之间的跃迁：相空间中每一点的跃迁维格纳函数的平方模是通过该点的反射的跃迁概率。粗粒化初始和最终能量提供了作为相空间路径积分的跃迁概率密度。它在这里调查在最简单的经典近似涉及微正则维格纳函数。反射算子生成一对能量壳之间的过渡，其概率密度由泊松括号的倒数模量在壳与其对的反射的交点上的积分给出。这对维格纳函数在其主焦散线处的奇异性在它们的交点上是很好的可积的，除了一个自由度。即使这种情况下是不直接相关的混沌系统的未来调查，它在这里示出如何改进的近似谱维格纳函数的艾里函数解决了奇异性。
英文摘要：
Phase space reflection operators lie at the core of the Wigner-Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): The square modulus of the transition Wigner function at each point in phase space is the transition probability for the reflection through that point. Coarsegraining the initial and final energies provides a transition probability density as a phase space path integral. It is here investigated in the simplest classical approximation involving microcanonical Wigner functions. A reflection operator generates a transition between a pair of energy shells with a probability density given by the integral of the inverse modulus of a Poisson bracket over the intersection of a shell with the reflection of its pair. The singularity of the pair of Wigner functions at their dominant caustics is nicely integrable over their intersection, except for a single degree of freedom. Even though this case is not directly relevant for future investigations of chaotic systems, it is shown here how the improved approximation of the spectral Wigner functions in terms of Airy functions resolves the singularity.
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