标题： 桌面上基于经典轨迹的半经典波函数。 能级量子化、疤痕与周期轨道 显示英文标题

标题： Semiclassical wave functions in billiards built on classical trajectories. Energy quantization, scars and periodic orbits

摘要： 提出了一种基于 Maslov-Fedoriuk 方法构造半经典波函数（SWF）的途径，这种方法似乎也适用于具有混沌经典极限的系统。考虑了一些被称为骨架的经典结构。骨架是对阿诺德环面的推广，能够涵盖混沌动力学。SWF 在构型空间中通过焦散奇点来延拓，而不是在相空间中，使用复时间方法。骨架表述为我们提供了一种新的半经典近似法算法，该算法被应用于构造 SWF，计算圆形和矩形台球的能量谱，以及构建 Bunimovich 台球的最简单 SWF 和相应的谱。讨论了疤痕现象及其由骨架方法描述的可能性。 PACS 编号：03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv 关键词： 薛定谔方程，半经典展开，拉格朗日流形，经典轨迹，混沌动力学，量子混沌，疤痕 显示英文摘要

摘要： A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schr\"odinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scars