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 
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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

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

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