TY - JOUR
T1 - Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices
AU - Cornean, Decebal Horia
AU - Garde, Henrik
AU - Støttrup, Benjamin
AU - Sørensen, Kasper Studsgaard
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Magnetic pseudodifferential operators
KW - Spectral estimates
KW - Generalized Hofstadter matrices
KW - Magnetic pseudodifferential operators
KW - Spectral estimates
KW - Generalized Hofstadter matrices
UR - http://www.scopus.com/inward/record.url?scp=85065132566&partnerID=8YFLogxK
U2 - 10.1007/s11868-018-0271-y
DO - 10.1007/s11868-018-0271-y
M3 - Journal article
SN - 1662-9981
VL - 10
SP - 307
EP - 336
JO - Journal of Pseudo-Differential Operators and Applications
JF - Journal of Pseudo-Differential Operators and Applications
IS - 2
ER -