Quantum dynamics and Gram's matrix

TL;DR

This paper introduces a method to analyze quantum dynamics by examining the eigenvalue spectrum of Gram matrices derived from vector sequences, revealing insights into quantum chaos and long-term behavior.

Contribution

It proposes a novel approach linking Gram matrix spectra to quantum chaoticity and long-time quantum dynamics analysis.

Findings

Eigenvalue distribution reflects quantum chaoticity.

Method applicable to long-time quantum system analysis.

Connects quantum and classical dynamics through spectral properties.

Abstract

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.