Quantum dynamics and Gram's Matrix
Quantum dynamics and Gram's Matrix
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DOI：
10.1209/epl/i2000-00163-6
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发表时间：
1999-01
期刊：
影响因子：
--
通讯作者：
M. Cock;M. Fannes;P. Spincemaille
中科院分区：
文献类型：
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作者：
M. Cock;M. Fannes;P. Spincemaille
我们建议使用相关联的格拉姆矩阵的频谱来分析长序列的向量的统计特性。这样的序列出现，例如，通过对哈密顿演化的频闪观测或通过在初始条件下被踢的量子动力学的重复作用。人们应该区分具有经典极限的动力系统和具有连续动力谱的无限量子系统。对于第一类，我们认为，当时间步长的数量，适当缩放相对于，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的极限。一方面，我们考虑具有可积经典极限的系统的极值模型。另一方面，一个随机系统模仿一个完全混沌的经典对应系统的量子动力学。对于无限大系统，格拉姆矩阵的本征值分布可以与众所周知的混沌特性（如正动力学熵）联系起来。
We propose to analyse the statistical properties of long sequences of vectors using the spectrum of the associated Gram matrix. Such sequences arise, e.g., by stroboscopic observation of a Hamiltonian evolution or by repeated action of a kicked quantum dynamics on an initial condition. One should distinguish between dynamical systems with a classical limit and infinite quantum systems with continuous dynamical spectrum. For the first class, we argue that when the number of time steps, suitably scaled with respect to , 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. On the one hand, we consider the extreme model of a system with integrable classical limit. A random system, on the other hand, mimics the quantum dynamics of a system with a completely chaotic classical counterpart. For infinite systems the eigenvalue distribution of the Gram matrix can be related to well-known characteristics of chaos such as positive dynamical entropy.