Mathematical Physics and Harmonic Analysis Seminar

Date: March 12, 2021

Time: 2:00PM - 3:00PM

Location: Zoom

Speaker: Jon Harrison, Baylor University

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

Abstract: Spectral statistics of classically chaotic quantum systems are often analyzed using their periodic orbit structure via Gutzwiller's trace formula, which holds in the semiclassical limit. We show that for chaotic 4-regular quantum graphs the variance of coefficients of the characteristic polynomial of the quantum evolution operator can be evaluated via periodic orbits without taking the semiclassical limit. The variance of the n-th coefficient is precisely determined by the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) with n edges that fall in the following classes: those with no self-intersections, and those where all the self-intersections consist of two sections of the pseudo orbit crossing at a single vertex (2-encounters of length zero).