Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

  • Horia D. Cornean
  • Henrik Garde
  • Benjamin Støttrup
  • Kasper S. Sørensen


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.


Magnetic pseudodifferential operators Spectral estimates Generalized Hofstadter matrices 

Mathematics Subject Classification

47A10 47G30 47G10 



H.C. gratefully acknowledges inspiring discussions with S. Beckus, J. Bellissard, B. Helffer, G. Nenciu, and R. Purice. This research is supported by grant 8021–00084B Mathematical Analysis of Effective Models and Critical Phenomena in Quantum Transport from The Danish Council for Independent Research | Natural Sciences.


  1. 1.
    Athmouni, N., Măntoiu, M., Purice, R.: On the continuity of spectra for families of magnetic pseudodifferential operators. J. Math. Phys. 51, Article ID 083517 (2010)Google Scholar
  2. 2.
    Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44(1), 45–57 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beckus, S., Bellissard, J.: Continuity of the spectrum of a field of self-adjoint operators. Ann. H. Poincaré 17(12), 3425–3442 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bellissard, J.: \(C^*\)-algebras in solid state physics: 2D electrons in uniform magnetic field. In: Evans, D., Takesaki, M. (eds.) Operator Algebras and Applications, London Mathematical Society Lecture Note Series 136, vol. 2, pp. 49–76. Cambridge University Press, Cambridge (1989)Google Scholar
  5. 5.
    Bellissard, J.: Lipshitz continuity of gap boundaries for Hofstadter-like spectra. Commun. Math. Phys. 160(3), 599–613 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bony, J.M.: Caractérisation des opérateurs pseudo-différentiels. Ecole Polytechnique, Séminaire E.D.P., Exposé no. XXIII (1996–1997)Google Scholar
  7. 7.
    Cornean, H.D.: On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schrödinger operators. Ann. H. Poincaré 11(5), 973–990 (2010)CrossRefGoogle Scholar
  8. 8.
    Cornean, H.D., Helffer, B., Purice, R.: Low lying spectral gaps induced by slowly varying magnetic fields. J. Funct. Anal. 273(1), 206–282 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cornean, H.D., Helffer, B., Purice, R.: Peierls’ substitution for low lying spectral energy windows. J. Spectr. Theory (2018) (to appear)Google Scholar
  10. 10.
    Cornean, H.D., Helffer, B., Purice, R.: A Beals criterion for magnetic pseudodifferential operators proved with magnetic Gabor frames. Commun. Part. Differ. Equ. (2018) (to appear)Google Scholar
  11. 11.
    Cornean, H.D., Purice, R.: On the Regularity of the Hausdorff Distance Between Spectra of Perturbed Magnetic Hamiltonians. In: Benguria, R., Friedman, E., Mantoiu, M. (eds.) Spectral Analysis of Quantum Hamiltonians, in Operator Theory: Advances and Applications, vol 224, pp 55–66. Springer, Basel (2012)Google Scholar
  12. 12.
    Cornean, H.D., Purice, R.: Spectral edge regularity of magnetic Hamiltonians. J. Lond. Math. Soc. 92(1), 89–104 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    De Nittis, G., Lein, M.: Applications of Magnetic \(\Psi \)DO Techniques to SAPT. Rev. Math. Phys. 23(3), 233–260 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Freund, S., Teufel, S.: Peierls substitution for magnetic Bloch bands. Anal. PDE 9(4), 773–811 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique). Mémoire de la SMF, No 34; Tome 116, Fasc.4 (1988)Google Scholar
  16. 16.
    Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper, LNP, vol. 345, pp. 118–197. Springer, Berlin (1989)zbMATHGoogle Scholar
  17. 17.
    Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14(6), 2239–2249 (1976)CrossRefGoogle Scholar
  18. 18.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, I, 2nd edn. Springer, New York (1990)zbMATHGoogle Scholar
  19. 19.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, III, 2nd edn. Springer, New York (1994)Google Scholar
  20. 20.
    Iftimie, V., Măntoiu, M., Purice, R.: Magnetic pseudodifferential operators. Publ. RIMS 43(3), 585–623 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Iftimie, V., Măntoiu, M., Purice, R.: Commutator criteria for magnetic pseudodifferential operators. Commun. Partial Differ. Equ. 35(6), 1058–1094 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Iftimie, V., Purice, R.: Magnetic Fourier integral operators. J. Pseudo-Differ. Oper. Appl. 2(2), 141–218 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Iftimie, V., Purice, R.: The Peierls–Onsager effective Hamiltonian in a complete gauge covariant setting: determining the spectrum. J. Spectr. Theory 5(3), 445–531 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Karasev, M.V., Osborn, T.A.: Sympletic areas, quantization and dynamics in electromagnetic fields. J. Math. Phys. 43, 756–788 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kato, T.: Pertubation Theory for Linear Operators, Reprint of 2nd edition. Springer, Berlin (1995)CrossRefGoogle Scholar
  26. 26.
    Luttinger, J.M.: The effect of a magnetic field on electrons in a periodic potential. Phys. Rev. 84(4), 814–817 (1951)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lein, M., Măntoiu, M., Richard, S.: Magnetic pseudodifferential operators with coefficients in \(C^*\)-algebras. Publ. Res. Inst. Math. Sci. 46(4), 755–788 (2010)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Măntoiu, M., Purice, R.: The Magnetic Weyl Calculus. J. Math. Phys. 45, 1394–1417 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Măntoiu, M., Purice, R., Richard, S.: Spectral and propagation results for magnetic Schrödinger operators; a \(C^*\)-algebraic approach. J. Funct. Anal. 250(1), 42–67 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nenciu, G.: Dynamics of Bloch electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians. Rev. Mod. Phys. 63(1), 91–127 (1991)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nenciu, G.: On asymptotic perturbation theory for quantum mechanics: almost invariant subspaces and gauge invariant magnetic perturbation theory. J. Math. Phys. 43(3), 1273–1298 (2002)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Nenciu, G.: On the Smoothness of Gap Boundaries for Generalized Harper Operators. In: Advances in Operator Algebras and Mathematical Physics. Theta Series in Advanced Mathematics, vol. 5, pp. 173–182. Theta, Bucharest (2005)Google Scholar
  33. 33.
    Panati, G., Spohn, H., Teufel, S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242(3), 547–578 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Peierls, R.E.: Quantum Theory of Solids. Oxford University Press, Oxford (1955)zbMATHGoogle Scholar
  35. 35.
    Sjöstrand, J.: Microlocal Analysis for the Periodic Magnetic Schrödinger Equation and Related Questions: CIME Lectures July 1989, in Microlocal Analysis and Applications, LNM, vol. 1495, pp. 237–332. Springer, Berlin (1991)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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