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Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

  • Horia D. Cornean
  • Henrik Garde
  • Benjamin Støttrup
  • Kasper S. Sørensen
Article

Abstract

First, we reconsider the magnetic pseudodifferential calculus and show that for a large class of non-decaying symbols, their corresponding magnetic pseudodifferential operators can be represented, up to a global gauge transform, as generalized Hofstadter-like, bounded matrices. As a by-product, we prove a Calderón–Vaillancourt type result. Second, we make use of this matrix representation and prove sharp results on the spectrum location when the magnetic field strength b varies. Namely, when the operators are self-adjoint, we show that their spectrum (as a set) is at least 1 / 2-Hölder continuous with respect to b in the Hausdorff distance. Third, when the magnetic perturbation comes from a constant magnetic field we show that their spectral edges are Lipschitz continuous in b. The same Lipschitz continuity holds true for spectral gap edges as long as the gaps do not close.

Keywords

Magnetic pseudodifferential operators Spectral estimates Generalized Hofstadter matrices 

Mathematics Subject Classification

47A10 47G30 47G10 

Notes

Acknowledgements

H.C. gratefully acknowledges inspiring discussions with S. Beckus, J. Bellissard, B. Helffer, G. Nenciu, and R. Purice. This research is 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.

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Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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