Asymptotics of Resonances Induced by Point Interactions

0484314 - ÚJF 2018 RIV PL eng J - Journal Article
Lipovský, J. - Lotoreichik, Vladimir
Asymptotics of Resonances Induced by Point Interactions.
Acta Physica Polonica A. Roč. 132, č. 6 (2017), s. 1677-1682. ISSN 0587-4246. E-ISSN 1898-794X
R&D Projects: GA ČR GA17-01706S
Institutional support: RVO:61389005
Keywords : self-adjoint three-dimensional Schrodinger operator * interactions * resonances
OECD category: Optics (including laser optics and quantum optics)
Impact factor: 0.857, year: 2017

We consider the resonances of the self-adjoint three-dimensional Schrodinger operator with point interactions of constant strength supported on the set X = {x(n)}(n-1)(N). The size of X is defined by V-X = max pi is an element of Pi(N) Sigma(N)(n = 1) vertical bar x(n) - x(pi(n))vertical bar, where Pi(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 is an element of R + O(1) as R -> infinity, where the constant W-X is an element of[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.
Permanent Link: http://hdl.handle.net/11104/0279447