Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

TL;DR

This paper develops a matrix representation for magnetic pseudodifferential operators, enabling precise spectral analysis and demonstrating continuity properties of spectral edges and gaps with respect to magnetic field strength.

Contribution

It introduces a generalized Hofstadter-like matrix representation for magnetic pseudodifferential operators, facilitating spectral analysis and continuity results.

Findings

Spectrum varies at least 1/2-Hölder continuously with magnetic field strength.

Spectral edges are Lipschitz continuous in magnetic field strength for constant fields.

Spectral gaps remain Lipschitz continuous as long as they do not close.

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 Calder\'on-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 $1/2$-H\"{o}lder 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.