Topological Resonances on Quantum Graphs

Yves Colin de Verdìère (1) , Francoise Truc (1)

1 IF - Institut Fourier

In this paper, we try to put the results of Smilansky and 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 exists compactly supported eigenfunctions. We give several estimateof the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case oftrees is also discussed.

Keywords: resonances; Quantum graphs; cycles in graphs

Domains: Mathematical Physics [math-ph]; Analysis of PDEs [math.AP]; Combinatorics [math.CO]; Spectral Theory [math.SP]

Yves Colin de Verdìère, Francoise Truc. Topological Resonances on Quantum Graphs. Annales Henri Poincaré, In press, ⟨10.1007/s00023-018-0672-8⟩. ⟨hal-01298494v2⟩