A trace formula for scattering resonances of unbalanced quantum graphs

TL;DR

This paper derives a trace formula connecting scattering resonances with integrals for unbalanced open quantum graphs, providing bounds on resonance counts and insights into spectral measure convergence.

Contribution

It introduces a new trace formula for unbalanced quantum graphs and establishes bounds and convergence results related to their resonances.

Findings

Derived a trace formula linking resonances and integrals

Established lower bounds on the number of resonances

Indicated convergence of spectral measures under Benjamini-Schramm limits

Abstract

Given an unbalanced open quantum graph, we derive a formula relating sums over its scattering resonances with integrals outside a strip. 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.