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.
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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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Kirsch, W., Krejčiřík, D. & Raikov, G. Lifshits Tails for Randomly Twisted Quantum Waveguides. J Stat Phys 171, 383–399 (2018). https://doi.org/10.1007/s10955-018-2001-5
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DOI: https://doi.org/10.1007/s10955-018-2001-5