Abstract

We present the probability distribution function of the nodal domain number for the right-isosceles triangle billiard. This is the first explicit result for a nonseparable plane polygonal billiard. The distribution function has a peak at \(2/\pi\) (the same as for integrable billiards) and also at \(2\sqrt{2}/\pi\). The only exact results previously known in this direction correspond to integrable billiards that are separable.

中文翻译：

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右等腰三角形台球节点域数的极限分布

摘要

我们提出了右等腰三角形台球的节点域数的概率分布函数。这是不可分离平面多边形台球的第一个显式结果。分布函数在\(2/\pi\)（与可积台球相同）和\(2\sqrt{2}/\pi\)处有一个峰值。在这个方向上先前已知的唯一确切结果对应于可分离的可积台球。