Lozej Črt, Bogomolny Eugene
<a href="https://ror.org/01bf9rw71">Max Planck Institute for the Physics of Complex Systems</a>, 01187 Dresden, Germany.
Université Paris-Saclay, CNRS, <a href="https://ror.org/00w67e447">LPTMS</a>, 91405 Orsay, France.
Phys Rev E. 2024 Aug;110(2-1):024213. doi: 10.1103/PhysRevE.110.024213.
Triangular billiards whose angles are rational multiples of π are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle, which permits one to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion, and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short-range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.
角度为π的有理倍数的三角台球是具有有趣经典和量子特性的伪可积模型的最简单示例之一。我们对八个量子化有理三角形的谱统计进行了广泛的数值研究，其中六个属于直角维奇三角形家族，两个是钝角有理三角形。为每个三角形计算了多达一百万个能级的大型谱样本，这使得人们能够非常精确地确定它们的谱统计。结果表明，它们属于中间类型，与混沌系统有一些共同特征，如能级排斥，与可积系统也有一些共同特征，如能级间距分布的指数尾部。中间谱统计的另一个显著特征是能级压缩性的有限值。我们详细分析了诸如能级间距分布等短程统计以及诸如数方差和谱形状因子等长程统计。数值数据与伽马分布模型之间显示出极好的一致性。
