On the Effective Size of a Non-Weyl Graph

TL;DR

This paper develops a method to analyze the resonance asymptotics of non-Weyl quantum graphs, reducing complexity and deriving bounds based on graph structure, with explicit resonance position calculations.

Contribution

It introduces a novel approach using pseudo orbit expansion to determine resonance asymptotics for non-Weyl graphs and reduces the graph complexity for analysis.

Findings

Derived bounds on resonance coefficients based on graph structure

Developed a method to reduce the number of edges in the directed graph

Explicitly identified positions of resolvent resonances

Abstract

We show how to find the coefficient by the leading term of the resonance asymptotics using the method of pseudo orbit expansion for quantum graphs which do not obey the Weyl asymptotics. For a non-Weyl graph we develop a method how to reduce the number of edges of a corresponding directed graph. Through this method we prove bounds on the above coefficient depending on the structure of the graph for graphs with the same lengths of internal edges. We explicitly find the positions of the resolvent resonances.