Non-Weyl resonance asymptotics for quantum graphs

Davies, E. B.; Pushnitski, Alexander

doi:10.2140/apde.2011.4.729

Cited by 35 publications

(86 citation statements)

References 32 publications

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“…Kottos and Smilansky proposed such a formula as a tool to study quantum chaos on graphs [KSm99]; the formula for graphs with a general vertex coupling has been derived in [BE09]. On infinite graphs high-energy behavior of resonances is of interest; it appears that for particular graph topologies and coupling conditions their distribution may not follow the usual Weyl law [DP11,DEL10,EL11].…”

Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

146

133

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More information about this series at http://www.springer.com/series/720The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals that the series is a suitable publication platform for both the mathematical and the theoretical physicist. The wider scope of the series is reflected by the composition of the editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research.

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Section: Notesmentioning

confidence: 99%

Quantum Waveguides

Exner

Kovařík

2015

Theoretical and Mathematical Physics

146

133

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“…If the bond lengths are changed continuously so do the positions of the poles of S(k). When bond lengths become rationally dependent some poles may move to the real axis indicating the appearance of a normalizable bound state embedded in the continuum (see [17][18][19]). …”

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confidence: 99%

Topological Resonances in Scattering on Networks (Graphs)

2013

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We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph -its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances 'topological' to emphasize their origin in the non-trivial connectivity. Topological resonances have a clear and unique signature which is apparent in the statistics of the resonance parameters (such as e.g., the width, the delay time or the wavefunction intensity in the graph). We discuss this phenomenon by providing analytical arguments supported by numerical simulation, and identify the features of the above distributions which depend on genuine topological quantities such as the length of the shortest cycle (girth). These signatures cannot be explained using any of the other paradigms for narrow resonances. Finally, we propose an experimental setting where the topological resonances could be demonstrated, and study the stability of the relevant distribution functions to moderate dissipation.

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“…On the other hand, it resembles the corresponding law for one-dimensional Schrödinger operators with regular potentials [11] and for quantum graphs [9,10].…”

Section: Discussionmentioning

confidence: 65%

Asymptotics of Resonances Induced by Point Interactions

Lipovský

Lotoreichik

2017

Acta Phys. Pol. A

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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 n=1 . The size of X is defined by VX = maxπ∈Π N N n=1 |xn−x π(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 linearwhere 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.

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Non-Weyl resonance asymptotics for quantum graphs

Cited by 35 publications

References 32 publications

Quantum Waveguides

Quantum Waveguides

Topological Resonances in Scattering on Networks (Graphs)

Asymptotics of Resonances Induced by Point Interactions

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