Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit

TL;DR

This paper evaluates a spectral statistic of chaotic quantum graphs directly from their periodic orbits without relying on the semiclassical limit, revealing connections between orbit structures and spectral properties.

Contribution

It introduces a method to compute a spectral statistic of quantum graphs solely from periodic orbits, bypassing the semiclassical approximation.

Findings

Variance of characteristic polynomial coefficients linked to orbit sets

Identifies the role of self-intersections in orbit contributions

Connects semiclassical results to total orbit counts

Abstract

Energy level statistics of quantized chaotic systems have been evaluated in the semiclassical limit via their periodic orbits using the Gutzwiller and related trace formulae. Here we evaluate a spectral statistic of chaotic 4-regular quantum graphs from their periodic orbits without the semiclassical limit. The variance of the n-th coefficient of the characteristic polynomial is determined by the sizes of the sets of distinct primitive periodic orbits with n bonds which have no self-intersections, and the sizes of the sets with a given number of self-intersections which all consist of two sections of the pseudo orbit crossing at a single vertex. Using this result we observe the mechanism that connects semiclassical results to the total number of orbits regardless of their structure.