## Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

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### DOI

• 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 language English Journal of Pseudo-Differential Operators and Applications 10 2 307-336 30 1662-9981 https://doi.org/10.1007/s11868-018-0271-y Published - 2019 Yes

### Research areas

• Generalized Hofstadter matrices, Magnetic pseudodifferential operators, Spectral estimates

Citationformats

ID: 196367531