Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices with angles with integer n. It is demonstrated that in the semiclassical limit the multiple singular scattering on such vertices, when optical boundaries of many scatters overlap, leads to the vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to the singular scattering on boundary vertices, waves propagated inside a periodic orbit pencil tend in the semiclassical limit to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems, the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes. For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.

多边形台球构成了一类特殊的模型。尽管它们的李雅普诺夫指数为零，但由于在具有整数n 的角度的奇异顶点上散射，它们涉及经典和量子特性. 结果表明，在半经典极限下，当许多散射的光学边界重叠时，这些顶点上的多重奇异散射会导致量子波函数沿由这些散射建立的直线消失。这种现象对于多边形台球具有特别重要的后果，其中周期性轨道（当它们存在时）形成平行射线束，从两侧受到奇异顶点的限制。由于边界顶点上的奇异散射，在周期轨道线形内传播的波沿线形边界在半经典极限内趋于零，从而形成弱相互作用的准模式。与混沌系统中的疤痕相反，所讨论的多边形台球中的准模式对于高激发态几乎是精确的，并且为简洁起见，它们被指定为超级疤痕。文中给出了许多具有明显超疤痕结构的三角台球和障碍台球的本征函数图片。特别关注开发检测和分析此类超级疤痕的定量方法。特别是，它证明了与固定周期轨道相关的超级疤痕波和障碍台球的特征函数之间的重叠是根据典型的弱相互作用准模式的 Breit-Wigner 分布分布的。对于称为 Veech 多边形的有理多边形台球的特殊子类，其中所有周期轨道都可以通过分析计算，在数值上论证和检查它们的特征函数在傅立叶空间中是分形的。