Vol. 4, No. 5, 2011

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Non-Weyl resonance asymptotics for quantum graphs

E. Brian Davies and Alexander Pushnitski

Vol. 4 (2011), No. 5, 729–756
Abstract

We consider the resonances of a quantum graph $\mathsc{G}$ 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 $\mathsc{G}$ in a disc of a large radius. We call $\mathsc{G}$ a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of $\mathsc{G}$. 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.

Keywords
quantum graph, resonance, Weyl asymptotics
Mathematical Subject Classification 2000
Primary: 34B45
Secondary: 35B34, 47E05