Improved eigenvalue sums for inferring quantum billiard geometry
Improved eigenvalue sums for inferring quantum billiard geometry
复制标题
改进的特征值和用于推断量子台球几何
DOI：
10.1088/0305-4470/20/9/026
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
M. Berry
中科院分区：
文献类型：
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作者：
M. Berry
描述了一种算法，用于获得封闭边界 B 的几何属性 Kj（例如其长度 L 或其中的面积 A），给定在 B 所界定的域上定义的某些波算子的最低 N 个特征值 (En)。该技术基于小 t 的配分函数的渐近展开： Phi (t)= Sigma n=1infinity exp(-Ent) 近似 t-1 Sigma j=0infinity Kjtj/2。采用四种不同的台球来说明该方法。第一个是矩形膜，它是经典可积的；对于其他三个，B 是非洲形状，这是典型的混乱：非洲膜、非洲阿哈罗诺夫-玻姆台球和非洲中微子（无质量狄拉克）台球。典型的结果是 A 和 L 可以从 125 个特征值重构为 104 个特征值中的几个部分（对于矩形，精度更高）。重建算法的效率似乎与 B 的经典混沌学无关。
An algorithm is described for obtaining successive approximations to geometric properties Kj of a closed boundary B (such as its length L or the area A within it), given the lowest N eigenvalues (En) of some wave operator defined on the domain bounded by B. The technique is based on the asymptotic expansion of the partition function for small t: Phi (t)= Sigma n=1infinity exp(-Ent) approximately t-1 Sigma j=0infinity Kjtj/2. Four different billiards are employed to illustrate the method. The first is the rectangular membrane, which is classically integrable; for the other three, B is an Africa shape, which is classically chaotic: Africa membrane, Africa Aharonov-Bohm billiard and Africa neutrino (massless Dirac) billiard. A typical result is that A and L can be reconstructed from 125 eigenvalues to a few parts in 104 (for the rectangle the accuracy is even higher). The efficiency of the reconstruction algorithm appears to be independent of the classical chaology of B.