Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

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.

arXiv comment20 pages

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Cited by in corpus (5)

• Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames
• A Calculus for Magnetic Pseudodifferential Super Operators
• From orbital magnetism to bulk-edge correspondence
• Sharp spectral stability for a class of singularly perturbed pseudo-differential operators
• Magnetic Pseudo-differential Operators with Hörmander Symbols Dominated by Tempered Weights