Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

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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.
Original languageEnglish
JournalJournal of Pseudo-Differential Operators and Applications
Volume10
Issue number2
Pages (from-to)307-336
Number of pages30
ISSN1662-9981
DOIs
Publication statusPublished - 2019

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Pseudodifferential Operators
Magnetic fields
Magnetic Field
Gages
Byproducts
Lipschitz Continuity
Hausdorff Distance
Spectral Gap
Matrix Representation
Lipschitz
Gauge
Calculus
Vary
Transform
Perturbation
Operator

Keywords

  • Magnetic pseudodifferential operators
  • Spectral estimates
  • Generalized Hofstadter matrices

Cite this

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title = "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 Calder{\'o}n-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.",
keywords = "Magnetic pseudodifferential operators, Spectral estimates, Generalized Hofstadter matrices, Magnetic pseudodifferential operators, Spectral estimates, Generalized Hofstadter matrices",
author = "Cornean, {Decebal Horia} and Henrik Garde and Benjamin St{\o}ttrup and S{\o}rensen, {Kasper Studsgaard}",
year = "2019",
doi = "10.1007/s11868-018-0271-y",
language = "English",
volume = "10",
pages = "307--336",
journal = "Journal of Pseudo-Differential Operators and Applications",
issn = "1662-9981",
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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

U2 - 10.1007/s11868-018-0271-y

DO - 10.1007/s11868-018-0271-y

M3 - Journal article

VL - 10

SP - 307

EP - 336

JO - Journal of Pseudo-Differential Operators and Applications

JF - Journal of Pseudo-Differential Operators and Applications

SN - 1662-9981

IS - 2

ER -