Non-Weyl asymptotics for quantum graphs with general coupling conditions

TL;DR

This paper investigates the resonance asymptotics of quantum graphs with various vertex coupling conditions, identifying when non-Weyl behavior occurs and providing insights into the special role of Kirchhoff conditions.

Contribution

It establishes criteria for non-Weyl asymptotics in quantum graphs with general couplings and extends analysis to graphs with nonuniform edge weights.

Findings

Non-Weyl asymptotics occur mainly under Kirchhoff or anti-Kirchhoff conditions for balanced vertices.

Graphs without permutation symmetry can exhibit diverse non-Weyl behaviors.

Insights into why Kirchhoff couplings are special from a resonance perspective.

Abstract

Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions, while for graphs without permutation numerous examples of non-Weyl behaviour can be constructed. Furthermore, we present an insight helping to understand what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance point of view. Finally, we demonstrate a generalization to quantum graphs with nonequal edge weights.