关于量子猫映射的矩阵元分布
On the distribution of matrix elements for the quantum cat map
摘要：
对于许多经典混沌系统，人们认为量子波函数是均匀分布的，即光滑观测量的矩阵元趋于观测量的相空间平均。本文研究去对称量子猫映射矩阵元的涨落。本文对正规化矩阵元的分布提出一个猜想，即它们的分布是SU（2）中独立矩阵迹的加权和的分布。这与一般的混沌系统相反，在一般的混沌系统中，预期分布是高斯分布。我们计算了归一化矩阵元素的二阶矩和四阶矩，并与我们的猜想一致。
英文摘要：
For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. In this paper we study the fluctuations of the matrix elements for the desymmetrized quantum cat map. We present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We compute the second and fourth moment of the normalized matrix elements and obtain agreement with our conjecture.
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