A trace formula for scattering resonances of unbalanced quantum graphs

Abstract

Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals on horizontal lines in the complex plane. We deduce lower bounds on the number of resonances (in bounded regions of the complex plane) that are independent of the size of the graph. We also deduce partial results indicating that Benjamini-Schramm convergence of open quantum graphs should imply convergence of the empirical spectral measures.