Non-Weyl resonance asymptotics for quantum graphs

Abstract

We consider the resonances of a quantum graph $\mathcal{G}$ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of $\mathcal{G}$ in a disc of a large radius. We call $\mathcal{G}$ a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of $\mathcal{G}$. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.