Quantum dynamics and Gram's matrix

Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven Celestijnenlaan 200D, B-3001 Heverlee, Belgium

Received: 14 May 1999
Accepted: 6 December 1999

Abstract

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.