acta physica slovaca

Acta Physica Slovaca 66, No.4, 265 – 363 (2016) (99 pages)


J. Lipovský
   Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czechia

Full text: ::pdf :: (Received 31 December 2016, accepted 23 February 2017)

Abstract: In the current review, we study the model of quantum graphs. We focus mainly on the resonance properties of quantum graphs. We define resolvent and scattering resonances and show their equivalence. We present various results on the asymptotics of the number of resolvent resonances in both non-magnetic and magnetic quantum graphs and find bounds on the coefficient by the leading term of the asymptotics. We explain methods how to find the spectral and resonance condition. Most of the notions and theorems are illustrated in examples. We show how to find resonances numerically and, in a simple example, we find trajectories of resonances in the complex plane. We discuss Fermi’s golden rule for quantum graphs and distribution of the mean intensity for the topological resonances.

PACS: 03.65.Ge, 03.65.Nk, 02.10.Ox
Keywords: Quantum graphs, Resonances, Weyl asymptotics, Scattering theory, Fermi’s golden rule
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