The Spectrum of the Billiard Laplacian of a Family of Random Billiards
The Spectrum of the Billiard Laplacian of a Family of Random Billiards
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随机台球族的台球拉普拉斯谱
DOI：
10.1007/s10955-010-0079-5
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发表时间：
2010
影响因子：
1.6
通讯作者：
Hong
中科院分区：
文献类型：
--
作者：
R. Feres;Hong
随机台球是台球动力学系统，其反射定律给出台球粒子的碰撞后方向作为碰撞前方向的函数，由马尔可夫（散射）算子P指定。微观结构又被定义为我们称之为台球室Q的东西，它的形状完全决定了算子P。这个算子，定义在一个适当的希尔伯特空间，是有界自伴的，对于这里考虑的例子，希尔伯特-施密特算子。这种随机台球的统计理论中的一个中心问题是将Q的几何特征和P的谱联系起来。我们表明，对于由尺度不变曲率K参数化的特定台球胞形状族，（图2），台球拉普拉斯算子P-I与普通球面拉普拉斯算子密切相关，并通过部分分析和部分数值方法表明，这是如何为小的K值提供关于P的谱的渐近信息的。它示出，特别是，散射关于入射角的二阶矩非常接近的光谱间隙的P。
Random billiards are billiard dynamical systems for which the reflection law giving the post-collision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P. Billiards with microstructure are random billiards whose Markov operator is derived from a “microscopic surface structure” on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cellQ, the shape of which completely determines the operator P. This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P. We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. 2), that the billiard Laplacian P−I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P.