论文标题
混合型系统中量子本特征的现象学：具有复杂相空间结构的柠檬台球
Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure
论文作者
论文摘要
柠檬台球的边界是由Heller和Tomsovic在Phys中介绍的两个相等单位半径与其中心之间的距离2B的相交来定义的。今天46 38（1993）。我们研究两个经典和量子柠檬台球，对于B = 0.1953，0.083，它们是具有复杂相位空间结构的混合型台球，没有明显的粘性区域。最近发表了对其光谱的初步研究（物理学1 1-14（2021））。我们计算出$ 10^6 $的连续特征状态及其庞加尔 - 霍西米（pH）功能，并通过研究熵定位措施和归一化的反参与率来分析其本地化属性。我们还引入了一个重叠指数，该指数测量了与经典和混乱区域的pH函数重叠程度。我们观察到与不变的托里和混乱状态相关的常规状态以及与经典的混乱区域以及混合型状态有关的存在。我们表明，根据浆果 - 罗布尼的图像和pH值均匀的半经典凝结原理，混合型状态的相对分数随着能量的增加而降低，因此在严格的半经典极限下，仅留下纯粹的规则和混乱状态。我们的方法为量子混合型汉密尔顿系统中pH功能的结构和定位特性提供了一般的现象学概述。
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 46 38 (1993). We study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently (Physics 1 1-14 (2021)). We calculate a great number $10^6$ of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We also introduce an overlap index which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions the relative fraction of mixed-type states decreases as a power law with increasing energy, thus in the strict semiclassical limit leaving only purely regular and chaotic states. Our approach offers a general very good phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.
