Spectral statistics and periodic orbits

TL;DR

This paper discusses semi-classical methods for calculating quantum spectral statistics in chaotic systems, showing their agreement with random matrix theory predictions and analyzing the zeros of the Riemann zeta function.

Contribution

It introduces multiple methods to connect spectral statistics of chaotic systems with random matrix theory, including the approach to the universal limit and detailed analysis of zeta zeros.

Findings

Spectral statistics match Gaussian Unitary Ensemble predictions.

Methods can describe the approach to universal spectral statistics.

Detailed analysis of Riemann zeta function zeros.

Abstract

The main purpose of these lectures is to discuss briefly recent methods of calculation of statistical properties of quantum eigenvalues for chaotic systems based on semi-classical trace formulas. Under the assumption that periodic orbit actions are non-commensurable it is demonstrated by a few different methods that the spectral statistics of chaotic systems without time-reversal invariance in the universal limit agrees with statistics of the Gaussian Unitary Ensemble of random matrices. The methods used permit to obtain not only the limiting statistics but also the way the spectral statistics of dynamical systems tends to the universal limit. The statistics of the Riemann zeta function zeros is considered in details.