经典量子混合型无粘性柠檬台球
Classical and quantum mixed-type lemon billiards without stickiness
摘要：
柠檬台球的边界由两个单位半径相等的圆的交点定义，其圆心之间的距离为$2B$，如Heller和Tomsovic在Phys. Today { \bf 46} 38（1993）中介绍的。本文是我们最近发表在Phys. Rev. E { \bf 103} 012204（2021）上的关于具有强粘性效应的经典和量子遍历柠檬台球（$B=0.5$）的论文的延续。在这里，我们研究了经典和量子柠檬台球，案例为$B=0.42,\;0.55,\; 0.6$，它们是没有粘性区域的混合型台球，因此可以作为具有简单划分相空间的系统的理想示例。经典的相图显示了一个大的混沌海的结构与均匀的混沌（无粘性区域）周围的一个大的规则的岛屿几乎没有进一步的子结构，完全覆盖不变的环面。乱海和规则岛屿的边界是平滑的，除了几个点。经典输运时间估计很短（只有几次碰撞），因此混沌本征态的局域化相当弱。在混沌区，量子态具有无粘性混合型系统的{ \em universal}性质：（i）利用Poincaré-Husimi（PH）函数将本征态分为规则态和混沌态。规则本征能服从泊松统计，而混沌本征能则服从Brody分布，其能级排斥指数$β$的值取决于混沌本征态局域化的强度. Berry-Robnik-Brody（BRB）分布很好地描述了整个光谱。
英文摘要：
The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance $2B$ between their centers, as introduced by Heller and Tomsovic in Phys. Today {\bf 46} 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard ($B=0.5$) with strong stickiness effects published in Phys. Rev. E {\bf 103} 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases $B=0.42,\;0.55,\; 0.6$, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following {\em universal} properties of mixed-type systems without stickiness in the chaotic regions:(i) Using the Poincaré-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent $β$, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution.
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