Semiclassical Asymptotics and Spectral Gaps for Periodic Magnetic Schrödinger Operators on Covering Manifolds

  • Yuri A. Kordyukov
Part of the Trends in Mathematics book series (TM)


We survey a method to prove the existence of gaps in the spectrum of periodic second-order elliptic partial differential operators, which was suggested by Kordyukov, Mathai and Shubin, and describe applications of this method to periodic magnetic Schrödinger operators on a Riemannian manifold, which is the universal covering of a compact manifold. We prove the existence of arbitrarily large number of gaps in the spectrum of these operators in the asymptotic limits of the strong electric field or the strong magnetic field under Morse type assumptions on the electromagnetic potential. We work on the level of spectral projections (and not just their traces) and obtain an asymptotic information about classes of these projections in K-theory. An important corollary is a vanishing theorem for the higher traces in cyclic cohomology for the spectral projections. This result is then applied to the quantum Hall effect.


Spectral Projection Hall Conductance Cyclic Cohomology Hermitian Connection Trivial Line Bundle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Bellissard, A. van Elst, H. Schulz-Baldes, The non-commutative geometry of the quantum Hall effect. J. Math. Phys. 35 (1994), 5373–5451.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    U. Bunke, On the gluing problem for the ŋ-invariant. J. Differential Geom. 41 (1995), 397–448.MATHMathSciNetGoogle Scholar
  3. [3]
    D. Burghelea, L. Friedlander, T. Kappeler, P. McDonald, Analytic and Reidemeister torsion for representations in finite type Hilbert modules. Geom. Funct. Anal. 6 (1996), 751–859.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    A. Carey, K. Hannabus, V. Mathai, P. McCann, Quantum Hall Effect on the hyperbolic plane. Comm. Math. Phys. 190 (1998), 629–673.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    A. Figotin, P. Kuchment, Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. I. Scalar model. SIAM J. Appl. Math. 56 (1996), 68–88. II. Twodimensional photonic crystals. SIAM J. Appl. Math. 56 (1996), 1561–1620.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    A. Figotin, P. Kuchment, Spectral properties of classical waves in high-contrast periodic media. SIAM J. Appl. Math. 58 (1998), 683–702.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    L. Friedlander, On the density of states of periodic media in the large coupling limit. Comm. Partial Differential Equations 27 (2002), 355–380.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Gromov, Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.MATHMathSciNetGoogle Scholar
  9. [9]
    B. Helffer, J. Sjöstrand, Multiple wells in the semiclassical limit. I. Comm. Partial Differential Equations 9 (1984), 337–408.MATHMathSciNetGoogle Scholar
  10. [10]
    B. Helffer, J. Sjöstrand, Effet tunnel pour l’équation de Schrödinger avec champ magnétique. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 625–657.MATHMathSciNetGoogle Scholar
  11. [11]
    B. Helffer, J. Sjöstrand, Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique). Mém. Soc. Math. France (N.S.) 34(1988).Google Scholar
  12. [12]
    B. Helffer, A. Mohamed, Caractérisation du spectre essential de l’opérateur de Schrödinger avec un champ magnétique. Ann. Inst. Fourier (Grenoble) 38 (1988), 95–112.MATHMathSciNetGoogle Scholar
  13. [13]
    B. Helffer, J. Sjöstrand, Analyse semi-classique pour l’équation de Harper. II. Comportement semi-classique près d’un rationnel. Mém. Soc. Math. France (N.S.) 40 (1990).Google Scholar
  14. [14]
    B. Helffer, J. Sjöstrand, Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum. Mém. Soc. Math. France (N.S.) 39 (1989), 1–124.Google Scholar
  15. [15]
    B. Helffer, J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper. In Schrödinger operators (Sønderborg, 1988), Lecture Notes in Phys., 345, Springer, 1989, pp. 118–197.MATHGoogle Scholar
  16. [16]
    B. Helffer, J. Sjöstrand, On diamagnetism and de Haas-van Alphen effect. Ann. Inst. H. Poincaré Phys. Théor. 52 (1990), 303–375.MATHGoogle Scholar
  17. [17]
    B. Helffer, A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells. J. Funct. Anal. 138 (1996), 40–81.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    B. Helffer, A. Morame, Magnetic bottles in connection with superconductivity, J. Funct. Anal. 185 (2001), 604–680.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    B. Helffer, A. Morame, Magnetic bottles for the Neumann problem: the case of dimension 3. Spectral and inverse spectral theory (Goa, 2000) Proc. Indian Acad. Sci. (Math. Sci.) 112 (2002), 71–84.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    B. Helffer, A. Morame, Magnetic bottles for the Neumann problem: Curvature effects in the case of dimension 3. Ann. Sci. Ec. Norm. Sup. 4 série 37 (2004), 105–170.MATHMathSciNetGoogle Scholar
  21. [21]
    B. Helffer, J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynõmes de champs de vecteurs. Birkhäuser, 1985.Google Scholar
  22. [22]
    R. Hempel, I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps. Comm. Math. Phys. 169 (1995), 237–259.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    R. Hempel, I. Herbst, Bands and gaps for periodic magnetic Hamiltonians. In: Operator Theory: Advances and Applications, Vol. 78. Birkhäuser, 1995, pp. 175–184.MATHMathSciNetGoogle Scholar
  24. [24]
    R. Hempel, K. Lienau, Spectral properties of periodic media in the large coupling limit. Comm. Partial Differential Equations 25 (2000), 1445–1470.MATHMathSciNetGoogle Scholar
  25. [25]
    R. Hempel, O. Post, Spectral gaps for periodic elliptic operators with high contrast: an overview. In: Progress in analysis, Vol. I, II (Berlin, 2001). World Sci. Publishing, 2003, pp. 577–587.MATHMathSciNetGoogle Scholar
  26. [26]
    I. Herbst, S. Nakamura, Schrödinger operators with strong magnetic fields: Quasi-periodicity of spectral orbits and topology. In: Differential operators and spectral theory. Amer. Math. Soc. Transl. Ser. 2, v. 189, Amer. Math. Soc., 1999, pp. 105–123.Google Scholar
  27. [27]
    Yu. A. Kordyukov, V. Mathai, M. Shubin, Equivalence of projections in semiclassical limit and a vanishing theorem for higher traces in K-theory. J. Reine Angew. Math. 581 (2005), 193–236.MATHMathSciNetGoogle Scholar
  28. [28]
    Yu. A. Kordyukov, Spectral gaps for periodic Schrödinger operators with strong magnetic fields. Comm. Math. Phys. 253 (2005), 371–384.MATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    M. Marcolli, V. Mathai, Twisted index theory for good orbifolds, II: fractional quantum numbers. Comm. Math. Phys. 217 (2001), 55–87.MATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    V. Mathai, M. Shubin, Semiclassical asymptotics and gaps in the spectra of magnetic Schrödinger operators. Geom. Dedicata 91 (2002), 155–173.MATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    S. Nakamura, J. Bellissard, Low energy bands do not contribute to the quantum Hall effect. Comm. Math. Phys. 131 (1990), 283–305.MATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    O. Post, Periodic manifolds with spectral gaps. J. Differential Equations 187 (2003), 23–45.MATHMathSciNetCrossRefGoogle Scholar
  33. [33]
    M. Shubin, Semiclassical asymptotics on covering manifolds and Morse Inequalities. Geom. Funct. Anal. 6 (1996), 370–409.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Yuri A. Kordyukov
    • 1
  1. 1.Institute of MathematicsRussian Academy of SciencesUfaRussia

Personalised recommendations