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Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

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  • Horia Cornean, Institut for Matematiske Fag, Aalborg Universitet, Denmark
  • Henrik Garde
  • Benjamin Støttrup, Institut for Matematiske Fag, Aalborg Universitet, Denmark
  • Kasper Studsgaard Sørensen, Institut for Matematiske Fag, Aalborg Universitet, Denmark
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.
Original languageEnglish
JournalJournal of Pseudo-Differential Operators and Applications
Pages (from-to)307-336
Number of pages30
Publication statusPublished - 2019
Externally publishedYes

    Research areas

  • Generalized Hofstadter matrices, Magnetic pseudodifferential operators, Spectral estimates

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