Scattering resonances of large weakly open quantum graphs

TL;DR

This paper studies the distribution of scattering resonances in large open quantum graphs, showing most are near the real axis and relate to eigenvalues of the corresponding closed graphs.

Contribution

It establishes the asymptotic distribution of resonances for large weakly open quantum graphs, linking them to the eigenvalues of the closed counterparts.

Findings

Most resonances are close to the real axis.

Resonance distribution matches the square-root of eigenvalues of closed graphs.

Results apply when the number of leads is small compared to edges.

Abstract

In this paper, we consider a sequence of open quantum graphs, with uniformly bounded data, and we are interested in the asymptotic distribution of their scattering resonances. Supposing that the number of leads in our quantum graphs is small compared to the total number of edges, we show that most resonances are close to the real axis. More precisely, the asymptotic distribution of resonances of our open quantum graphs is the same as the asymptotic distribution of the square-root of the eigenvalues of the closed quantum graphs obtained by removing all the leads.