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Non-Weyl asymptotics for quantum graphs with general coupling conditions

E Brian Davies Pavel Exner Ji Lipovsk

TBD mathscidoc:1910.43081

Journal of Physics A: Mathematical and Theoretical, 43, (47), 474013, 2010.11
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 the case of Kirchhoff or anti-Kirchhoff conditions. While for graphs without permutation symmetry numerous examples of non-Weyl behaviour can be constructed. Furthermore, we present an insight into what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance point of view. Finally, we demonstrate a generalization to quantum graphs with unequal edge weights.
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@inproceedings{e2010non-weyl,
  title={Non-Weyl asymptotics for quantum graphs with general coupling conditions},
  author={E Brian Davies, Pavel Exner, and Ji Lipovsk},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020125237073477610},
  booktitle={Journal of Physics A: Mathematical and Theoretical},
  volume={43},
  number={47},
  pages={474013},
  year={2010},
}
E Brian Davies, Pavel Exner, and Ji Lipovsk. Non-Weyl asymptotics for quantum graphs with general coupling conditions. 2010. Vol. 43. In Journal of Physics A: Mathematical and Theoretical. pp.474013. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020125237073477610.
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