Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

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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
Issue number2
Pages (from-to)307-336
Number of pages30
Publication statusPublished - 2019



  • Magnetic pseudodifferential operators
  • Spectral estimates
  • Generalized Hofstadter matrices

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