Communications in Mathematical Physics

, Volume 253, Issue 2, pp 371–384

Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields

Article

Abstract

We consider Schrödinger operators Hh=(ihd+A)*(ihd+A) with the periodic magnetic field B=dA on covering spaces of compact manifolds. Using methods of a paper by Kordyukov, Mathai and Shubin [14], we prove that, under some assumptions on B, there are in arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of the strong magnetic field h→0.

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References

  1. 1.
    Figotin, A., Kuchment, P.: Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. I. Scalar model. SIAM J. Appl. Math. 56, 68–88 (1996); II. Two-dimensional photonic crystals. SIAM J. Appl. Math. 56, 1561–1620 (1996)MATHGoogle Scholar
  2. 2.
    Figotin, A., Kuchment, P.: Spectral properties of classical waves in high-contrast periodic media. SIAM J. Appl. Math. 58, 683–702 (1998)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Friedlander, L.: On the density of states of periodic media in the large coupling limit. Commun. Partial Differ. Eqs. 27, 355–380 (2002)CrossRefMATHGoogle Scholar
  4. 4.
    Helffer, B., Mohamed, A.: Caractérisation du spectre essential de l’opérateur de Schrödinger avec un champ magnétique. Ann. Inst. Fourier (Grenoble) 38, 95–112 (1988)MathSciNetMATHGoogle Scholar
  5. 5.
    Helffer, B., Mohamed, A.: Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells. J. Funct. Anal. 138, 40–81 (1996)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185, 604–680 (2001)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Helffer, B., Morame, A.: Magnetic bottles for the Neumann problem: the case of dimension 3. In: Spectral and inverse spectral theory (Goa, 2000). Proc. Indian Acad. Sci. (Math. Sci.) 112, 71–84 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Helffer, B., Morame, A.: Magnetic bottles for the Neumann problem: Curvature effects in the case of dimension 3. To appear in Ann. Sci. Ecole Norm. Sup. 2004Google Scholar
  9. 9.
    Helffer, B., Nourrigat, J.: Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs. Boston: Birkhäuser, 1985Google Scholar
  10. 10.
    Hempel, R., Herbst, I.: Strong magnetic fields, Dirichlet boundaries, and spectral gaps. Comm. Math. Phys. 169, 237–259 (1995)MathSciNetMATHGoogle Scholar
  11. 11.
    Hempel, R., Lienau, K.: Spectral properties of periodic media in the large coupling limit. Comm. Partial Differ. Eqs. 25, 1445–1470 (2000)MATHGoogle Scholar
  12. 12.
    Hempel, R., Post, O.: Spectral gaps for periodic elliptic operators with high contrast: an overview. http://arxiv.org/abs/math-ph/0207020, 2002
  13. 13.
    Herbst, I., Nakamura, S.: Schrödinger operators with strong magnetic fields: Quasi-periodicity of spectral orbits and topology. In: Differential operators and spectral theory. Am. Math. Soc. Transl. Ser. 2, v. 189, Providence RI: Am. Math. Soc., 1999, pp. 105 – 123Google Scholar
  14. 14.
    Kordyukov, Yu. A., Mathai, V., Shubin, M.: Equivalence of projections in semiclassical limit and a vanishing theorem for higher traces in K-theory. http://arxiv.org/abs/math.DG/0305189, 2003, to appear in J. Reine Angew. Math.
  15. 15.
    Mathai, V., Shubin, M.: Semiclassical asymptotics and gaps in the spectra of magnetic Schrödinger operators. Geom. Dedicata 91, 155–173 (2002)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Shubin, M.: Discrete magnetic Laplacian. Commun. Math. Phys. 164, 259–275 (1994)MathSciNetMATHGoogle Scholar
  17. 17.
    Shubin, M.: Semiclassical asymptotics on covering manifolds and Morse Inequalities. Geom. Anal. Funct. Anal. 6, 370–409 (1996)MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institute of MathematicsRussian Academy of SciencesUfaRussia

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