Asymptotics of Resonances Induced by Point Interactionsx

J. Lipovskýa and V. Lotoreichik b aDepartment of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 50003 Hradec Králové, Czechia bDepartment of Theoretical Physics, Nuclear Physics Institute CAS, 25068 Řež near Prague, Czechia

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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=1N. The size of X is defined by VX=maxπ∊ ΠN ∑n=1N |xn-xπ(n)|, where ΠN 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 (WX/π)R + O(1) as R → ∞, where the constant WX ∈ [0,VX] can be seen as the effective size of X. Moreover, we show that there exist a configuration of any number of points such that WX=VX. Finally, we construct an example for N=4 with WX < VX, which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.

DOI: 10.12693/APhysPolA.132.1677 PACS numbers: 03.65.Ge, 03.65.Nk, 02.10.Ox