Two-point spectral correlations for the square billiard
Two-point spectral correlations for the square billiard
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DOI：
10.1088/0305-4470/30/6/009
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发表时间：
1997-03
期刊：
影响因子：
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通讯作者：
R. Connors;J. Keating
中科院分区：
文献类型：
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作者：
R. Connors;J. Keating
研究了方台球量子谱中的两点相关。这个系统的不寻常之处在于，能级的简并在半经典极限中以这样一种方式增加，使得平均能级分离不是由平均态密度的倒数给出的。因此，例如，标准水平间距分布并不趋向于可积系统所期望的泊松极限。本文利用概率技术和经典数论相结合的方法计算了一类退化加权两点相关函数的首阶渐近形式。结果显示了关于平均值的数论波动，平均值是两项的和：一项具有通常的（恒定的）泊松形式，第二项表示一个小的校正，它作为相关距离的倒数而衰减。数值计算证实了这一点。
We investigate the two-point correlations in the quantum spectrum of the square billiard. This system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states. Hence, for example, the standard level spacings distribution does not tend to the Poissonian limit expected for integrable systems. In this paper we calculate the leading-order asymptotic form of a degeneracy-weighted two-point correlation function using a combination of probabilistic techniques and classical number theory. The result exhibits number-theoretical fluctuations about a mean which is a sum of two terms: one having the usual (constant) Poissonian form and the second representing a small correction which decays as the inverse of the correlation distance. This is confirmed by numerical computations.