E. Brian Davies, Pavel Exner, Jiri Lipovsky
Non-Weyl asymptotics for quantum graphs with general coupling conditions
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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.