Hölder Continuity of the Spectra for Aperiodic Hamiltonians

Siegfried Beckus, Jean Bellissard, Horia Cornean

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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.

Original languageEnglish
JournalAnnales Henri Poincare
Volume20
Issue number11
Pages (from-to)3603-3631
Number of pages29
ISSN1424-0637
DOIs
Publication statusPublished - 1 Nov 2019

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Keywords

  • spectrum location
  • aperiodic Hamiltonians
  • operator theory

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