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Pseudo-Orbit Expansion for the Resonance Condition on Quantum Graphs and the Resonance Asymptotics

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Lipovský J.

Acta Physica Polonica A

2015

vol. 128

issue 6

968-973

In this note we explain the method how to find the resonance condition on quantum graphs, which is called pseudo-orbit expansion. In three examples with standard coupling we show in detail how to obtain the resonance condition. We focus on non-Weyl graphs, i.e. the graphs which have fewer resonances than expected. For these graphs we explain benefits of the method of "deleting edges" for simplifying the graph.

Asymptotics of Resonances Induced by Point Interactions

63%

Lipovský J., Lotoreichik V.

Acta Physica Polonica A

2017

vol. 132

issue 6

1677-1682

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.

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2 Acta Physica Polonica A

1 2017

1 2015

2 Lipovský J.

1 Lotoreichik V.

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Pseudo-Orbit Expansion for the Resonance Condition on Quantum Graphs and the Resonance Asymptotics

Asymptotics of Resonances Induced by Point Interactions