量子动力学和格拉姆矩阵
Quantum dynamics and Gram's matrix
摘要：
我们建议分析的统计性质的一系列向量使用的频谱相关联的革兰氏矩阵。这样的序列例如通过确定性踢量子动力学在初始条件下的重复作用或通过随机过程而产生。我们认为，当时间步长的数量，适当缩放相对于$\hbar$，增加，限制的特征值分布的革兰氏矩阵反映了可能的量子混沌的原始系统，因为它往往是其经典极限。随后，这一想法被应用到研究的时间尺度上的可用状态的希尔伯特空间的维度的随机向量序列的长时间属性。
英文摘要：
We propose to analyse the statistical properties of a sequence of vectors using the spectrum of the associated Gram matrix. Such sequences arise e.g. by the repeated action of a deterministic kicked quantum dynamics on an initial condition or by a random process. We argue that, when the number of time-steps, suitably scaled with respect to $\hbar$, increases, the limiting eigenvalue distribution of the Gram matrix reflects the possible quantum chaoticity of the original system as it tends to its classical limit. This idea is subsequently applied to study the long-time properties of sequences of random vectors at the time scale of the dimension of the Hilbert space of available states.
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