关于量子猫映射矩阵元的涨落
On the fluctuations of matrix elements of the quantum cat map
摘要：
研究了量子猫映射的对角矩阵元关于其极限的涨落。我们表明，经过适当的规范化，第五次中心时刻的Hecke基础消失在半经典极限，证实了部分猜想的Kurlberg和Rudnick。我们还研究了短窗口中矩阵元素的和。对于具有零均值的观测量，这些和的一阶矩为零，并且方差由作者与Kurlberg和Rudnick确定。我们表明，如果窗口是足够小的普朗克常数，第三时刻消失，如果我们正常化，使方差是一阶。
英文摘要：
We study the fluctuations of the diagonal matrix elements of the quantum cat map about their limit. We show that after suitable normalization, the fifth centered moment for the Hecke basis vanishes in the semiclassical limit, confirming in part a conjecture of Kurlberg and Rudnick. We also study sums of matrix elements lying in short windows. For observables with zero mean, the first moment of these sums is zero, and the variance was determined by the author with Kurlberg and Rudnick. We show that if the window is sufficiently small in terms of Planck's constant, the third moment vanishes if we normalize so that the variance is of order one.
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