Asymptotics of Resonances Induced by Point Interactions

• Authors: J. Lipovský , V. Lotoreichik
• Languages of publication: EN

Abstracts

EN

We consider the resonances of the self-adjoint three-dimensional Schrödinger operator with point interactions of constant strength supported on the set X={xₙ}_{n=1}^{N}. The size of X is defined by V_{X} = max_{π ∈ Π_{N}} ∑_{n=1}^{N} |xₙ - x_{π(n)}|, where Π_{N} is the family of all the permutations of the set {1,2,...,N}. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius R behaves asymptotically linear W_{X}/πR + O(1) as R → ∞, where the constant W_{X} ∈ [0,V_{X}] can be seen as the effective size of X. Moreover, we show that there exist a configuration of any number of points such that W_{X}=V_{X}. Finally, we construct an example for N=4 with W_{X} < V_{X}, which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.

Keywords

EN

03.65.Ge; 03.65.Nk; 02.10.Ox

Discipline

• 02.10.Ox: Combinatorics; graph theory
• 03.65.Nk: Scattering theory
• 03.65.Ge: Solutions of wave equations: bound states

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2017
• Volume: 132
• Issue: 6
• Pages: 1677-1682

Dates

published

2017-12

Contributors

author

J. Lipovský

• Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 50003 Hradec Králové, Czechia

author

V. Lotoreichik

• Department of Theoretical Physics, Nuclear Physics Institute CAS, 25068 Řež near Prague, Czechia

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Identifiers

DOI

10.12693/APhysPolA.132.1677