Spectral Statistics: From Disordered to Chaotic Systems

TL;DR

This paper explores the connection between disordered and chaotic systems by linking their operators, allowing the extension of diffusive regime results to general chaotic systems, and analyzing spectral statistics around the Heisenberg time.

Contribution

It introduces a method to relate disordered and chaotic systems via operator identification, enabling broader analysis of spectral statistics.

Findings

Derived the two-point level density correlator for chaotic systems.

Calculated the structure factor and its behavior near the Heisenberg time.

Quantitatively linked spectral features to short periodic orbits.

Abstract

The relation between disordered and chaotic systems is investigated. It is obtained by identifying the diffusion operator of the disordered systems with the Perron-Frobenius operator in the general case. This association enables us to extend results obtained in the diffusive regime to general chaotic systems. In particular, the two--point level density correlator and the structure factor for general chaotic systems are calculated and characterized. The behavior of the structure factor around the Heisenberg time is quantitatively described in terms of short periodic orbits.