Journal of Statistical Physics

, Volume 171, Issue 3, pp 383–399 | Cite as

Lifshits Tails for Randomly Twisted Quantum Waveguides

Article
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Abstract

We consider the Dirichlet Laplacian \(H_\gamma \) on a 3D twisted waveguide with random Anderson-type twisting \(\gamma \). We introduce the integrated density of states \(N_\gamma \) for the operator \(H_\gamma \), and investigate the Lifshits tails of \(N_\gamma \), i.e. the asymptotic behavior of \(N_\gamma (E)\) as \(E \downarrow \inf \mathrm{supp}\, dN_\gamma \). In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.

Keywords

Randomly twisted quantum waveguide Dirichlet Laplacian Integrated density of states Lifshits tails 

Mathematics Subject Classification

82B44 35R60 47B80 81Q10 

Notes

Acknowledgements

The authors gratefully acknowledge the partial support of the Chilean Scientific Foundation Fondecyt under Grants 1130591 and 1170816. D. Krejčiřík was also partially supported by the GACR Grant No. 18-08835S and by FCT (Portugal) through Project PTDC/MAT-CAL/4334/2014. A considerable part of this work has been done during W. Kirsch’s visits to the Pontificia Universidad Católica de Chile in 2015 and 2016. He thanks this university for hospitality. Another substantial part of this work has been done during G. Raikov’s visits to the University of Hagen, Germany, the Czech Academy of Sciences, Prague, and the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia. He thanks these institutions for financial support and hospitality.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Werner Kirsch
    • 1
  • David Krejčiřík
    • 2
  • Georgi Raikov
    • 3
  1. 1.Fakultät für Mathematik und InformatikFernUniversität in HagenHagenGermany
  2. 2.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czech Republic
  3. 3.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiago de ChileChile

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