On the multilevel internal structure of the asymptotic distribution of resonances

We prove that the set of resonances Σ(H) has a multilevel asymptotic structure for the following classes of Hamiltonians H: Schrödinger operators with point interactions yj∈R3, quantum graphs, and 1-D photonic crystals. In the case of N≥2 point interactions, Σ(H) consists of a finite number of seque...

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Bibliographic Details
Published in:Journal of Differential Equations 2019-11, Vol.267 (11), p.6171-6197
Main Authors: Albeverio, Sergio, Karabash, Illya M.
Format: Article
Language:English
Subjects:
Counting function of narrow resonances
Delta-interaction
Density of resonances
Exponential polynomial
Scattering pole
Topological resonance
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Summary:We prove that the set of resonances Σ(H) has a multilevel asymptotic structure for the following classes of Hamiltonians H: Schrödinger operators with point interactions yj∈R3, quantum graphs, and 1-D photonic crystals. In the case of N≥2 point interactions, Σ(H) consists of a finite number of sequences with logarithmic asymptotics. Leading parameters μn of these sequences are connected with the geometry of the family Y={yj}j=1N. The minimal parameter μmin corresponds to sequences with ‘more narrow’ and so more observable resonances. The asymptotic density of narrow resonances is expressed via the multiplicity of μmin and is connected with symmetries of Y. In the cases of quantum graphs and 1-D photonic crystals, the decomposition of Σ(H) into asymptotic sequences is proved under commensurability conditions. To address the general quantum graph, we introduce asymptotic density functions for two classes of complex strips. The obtained results and effects are compared with obstacle scattering.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2019.06.020