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Hölder Continuity of the Spectra for Aperiodic Hamiltonians

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Abstract

We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are “close to each other” if, up to a translation, they “almost coincide” on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.

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Acknowledgements

This research was supported through the program “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2018. This research has been supported by Grant 8021–00084B Mathematical Analysis of Effective Models and Critical Phenomena in Quantum Transport from The Danish Council for Independent Research Natural Sciences. J.B. thanks the School of Mathematics at the Georgia Institute of Technology and the Fachbereich Mathematik at the Westfälische Wilhelms-Universität Münster.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Potsdam, Potsdam, Germany

    Siegfried Beckus

  2. Fachbereich 10 Mathematik und Informatik, Westfälische Wilhelms-Universität, Einsteinstrasse 62, 48149, Münster, Germany

    Jean Bellissard

  3. Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220, Aalborg, Denmark

    Horia Cornean

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  1. Siegfried Beckus
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  2. Jean Bellissard
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  3. Horia Cornean
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Correspondence to Siegfried Beckus.

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Communicated by Jan Derezinski.

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Beckus, S., Bellissard, J. & Cornean, H. Hölder Continuity of the Spectra for Aperiodic Hamiltonians. Ann. Henri Poincaré 20, 3603–3631 (2019). https://doi.org/10.1007/s00023-019-00848-6

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