Semiclassical calculation of spectral correlation functions of chaotic systems

TL;DR

This paper develops a semiclassical method to compute spectral correlation functions in chaotic quantum systems, confirming universality predictions from random matrix theory across different symmetry classes.

Contribution

It introduces a semiclassical framework for n-point spectral correlations in chaotic systems, explicitly calculating leading order results for unitary and orthogonal classes.

Findings

Results match random matrix theory predictions

Supports the universality conjecture in spectral statistics

Provides explicit calculations for the first correlation functions

Abstract

We present a semiclassical approach to n-point spectral correlation functions of quantum systems whose classical dynamics is chaotic, for arbitrary n. The basic ingredients are sets of periodic orbits that have nearly the same action and therefore provide constructive interference. We calculate explicitly the first correlation functions, to leading orders in their energy arguments, for both unitary and orthogonal symmetry classes. The results agree with corresponding predictions from random matrix theory, thereby giving solid support to the conjecture of universality.