Jon Harrison

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

Energy level statistics of quantized chaotic systems are often evaluated in the semiclassical limit via their periodic orbits using the Gutzwiller or related trace formulae. Here, we evaluate a spectral statistic of 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 crossing at a single vertex. Using this we observe the mechanism that connects semiclassical results to the total number of orbits regardless of their structure.