Topological Resonances on Quantum Graphs

@article{ColindeVerdire2016TopologicalRO,
  title={Topological Resonances on Quantum Graphs},
  author={Yves Colin de Verdi{\`e}re and Françoise Truc},
  journal={Annales Henri Poincar{\'e}},
  year={2016},
  volume={19},
  pages={1419-1438}
}
In this paper, we try to put the results of Smilansky et al. on “Topological resonances” on a mathematical basis. A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exist compactly supported eigenfunctions. We give several estimates on the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case of trees is also discussed. 

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