Pseudo orbit expansion for the resonance condition on quantum graphs and the resonance asymptotics

TL;DR

This paper introduces a pseudo orbit expansion method to determine resonance conditions on quantum graphs, demonstrating its effectiveness especially for non-Weyl graphs with fewer resonances than expected.

Contribution

It presents a detailed method for deriving resonance conditions on quantum graphs using pseudo orbit expansion, including a technique for simplifying non-Weyl graphs by deleting edges.

Findings

The pseudo orbit expansion method effectively finds resonance conditions.

The method simplifies analysis of non-Weyl graphs.

Edge deletion technique aids in resonance computation.

Abstract

In this note we explain the method how to find the resonance condition on quantum graphs, which is called pseudo orbit expansion. In three examples with standard coupling we show in detail how to obtain the resonance condition. We focus on non-Weyl graphs, i.e. the graphs which have fewer resonances than expected. For these graphs we explain benefits of the method of "deleting edges" for simplifying the graph.