Periodic magnetic Schrödinger operators: Spectral gaps and tunneling effect

Article
  • 39 Downloads

Abstract

A periodic Schrödinger operator on a noncompact Riemannian manifold M such that H1(M, ℝ) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group is considered. Under some additional conditions on the magnetic field, the existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit is established. The proofs are based on the study of the tunneling effect in the corresponding quantum system.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations (Princeton Univ. Press, Princeton, NJ, 1982), Math. Notes 29.MATHGoogle Scholar
  2. 2.
    J. Brüning, S. Yu. Dobrokhotov, and K. V. Pankrashkin, “The Spectral Asymptotics of the Two-Dimensional Schrödinger Operator with a Strong Magnetic Field. I, II,” Russ. J. Math. Phys. 9(1), 14–49 (2002); 9 (4), 400–416 (2002); arXiv:math-ph/0411012.MATHMathSciNetGoogle Scholar
  3. 3.
    U. Carlsson, “An Infinite Number of Wells in the Semi-classical Limit,” Asymptotic Anal. 3(3), 189–214 (1990).MATHMathSciNetGoogle Scholar
  4. 4.
    M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-classical Limit (Cambridge Univ. Press, Cambridge, 1999), LMS Lect. Note Ser. 268.MATHGoogle Scholar
  5. 5.
    R. L. Frank, “On the Tunneling Effect for Magnetic Schrödinger Operators in Antidot Lattices,” Asymptotic Anal. 48(1–2), 91–120 (2006).Google Scholar
  6. 6.
    B. Helffer, “Introduction to Semi-classical Methods for the Schrödinger Operator with Magnetic Fields,” in Aspects théoriques et appliqués de quelques EDP issues de la géométrie ou de la physique: Proc. CIMPA School, Damas (Syrie), 2004 (Soc. Math. France, Paris), Séminaires et Congrès (in press).Google Scholar
  7. 7.
    B. Helffer and Yu. A. Kordyukov, “Semiclassical Asymptotics and Gaps in the Spectra of Periodic Schrödinger Operators with Magnetic Wells,” Trans. Am. Math. Soc. 360, 1681–1694 (2008).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    B. Helffer and A. Mohamed, “Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique,” Ann. Inst. Fourier 38, 95–112 (1988).MATHMathSciNetGoogle Scholar
  9. 9.
    B. Helffer and A. Mohamed, “Semiclassical Analysis for the Ground State Energy of a Schrödinger Operator with Magnetic Wells,” J. Funct. Anal. 138, 40–81 (1996).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    B. Helffer and A. Morame, “Magnetic Bottles in Connection with Superconductivity,” J. Funct. Anal. 185, 604–680 (2001).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. Helffer and J. Nourrigat, Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs (Birkhäuser, Boston, 1985).MATHGoogle Scholar
  12. 12.
    B. Helffer and J. Sjöstrand, “Multiple Wells in the Semi-classical Limit. I,” Commun. Part. Diff. Eqns. 9, 337–408 (1984).MATHCrossRefGoogle Scholar
  13. 13.
    B. Helffer and J. Sjöstrand, “Puits multiples en limite semi-classique. II: Interaction moléculaire. Symétries. Perturbation,” Ann. Inst. H. Poincaré, Phys. Théor. 42(2), 127–212 (1985).MATHGoogle Scholar
  14. 14.
    B. Helffer and J. Sjöstrand, “Effet tunnel pour l’équation de Schrödinger avec champ magnétique,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. 4, 14, 625–657 (1987).MATHGoogle Scholar
  15. 15.
    B. Helffer and J. Sjöstrand, Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique) (Soc. Math. France, Paris, 1988), Mém. Soc. Math. France (N.S.) 34.Google Scholar
  16. 16.
    B. Helffer and J. Sjöstrand, “Équation de Schrödinger avec champ magnétique et équation de Harper,” in Schrödinger operators, Sønderborg, 1988 (Springer, Berlin, 1989), Lect. Notes Phys. 345, pp. 118–197.CrossRefGoogle Scholar
  17. 17.
    R. Hempel and I. Herbst, “Strong Magnetic Fields, Dirichlet Boundaries, and Spectral Gaps,” Commun. Math. Phys. 169, 237–259 (1995).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Hempel and O. Post, “Spectral Gaps for Periodic Elliptic Operators with High Contrast: An Overview,” in Progress in Analysis: Proc. 3rd ISAAC Congr., Berlin, 2001 (World Sci., River Edge, NJ, 2003), Vol. 2, pp. 577–587.Google Scholar
  19. 19.
    I. Herbst and S. Nakamura, “Schrödinger Operators with Strong Magnetic Fields: Quasi-periodicity of Spectral Orbits and Topology,” in Differential Operators and Spectral Theory (Am. Math. Soc., Providence, RI, 1999), AMS Transl., Ser. 2, 189, pp. 105–123.Google Scholar
  20. 20.
    Yu. Kordyukov, V. Mathai, and M. Shubin, “Equivalence of Spectral Projections in Semiclassical Limit and a Vanishing Theorem for Higher Traces in K-Theory,” J. Reine Angew. Math. 581, 193–236 (2005).MATHMathSciNetGoogle Scholar
  21. 21.
    Yu. A. Kordyukov, “Spectral Gaps for Periodic Schrödinger Operators with Strong Magnetic Fields,” Commun. Math. Phys. 253, 371–384 (2005).MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Yu. A. Kordyukov, “Semiclassical Asymptotics and Spectral Gaps for Periodic Magnetic Schrödinger Operators on Covering Manifolds,” in C*-Algebras and Elliptic Theory (Birkhäuser, Basel, 2006), Trends in Mathematics, pp. 129–150.CrossRefGoogle Scholar
  23. 23.
    V. Mathai and M. Shubin, “Semiclassical Asymptotics and Gaps in the Spectra of Magnetic Schrödinger Operators,” Geom. Dedicata 91, 155–173 (2002).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    R. Montgomery, “Hearing the Zero Locus of a Magnetic Field,” Commun. Math. Phys. 168, 651–675 (1995).MATHCrossRefGoogle Scholar
  25. 25.
    S. Nakamura, “Band Spectrum for Schrödinger Operators with Strong Periodic Magnetic Fields,” in Partial Differential Operators and Mathematical Physics, Holzhau, 1994 (Birkhäuser, Basel, 1995), Oper. Theory: Adv. Appl. 78, pp. 261–270.Google Scholar
  26. 26.
    X.-B. Pan and K.-H. Kwek, “Schrödinger Operators with Non-degenerately Vanishing Magnetic Fields in Bounded Domains,” Trans. Am. Math. Soc. 354, 4201–4227 (2002).MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Institute of MathematicsRussian Academy of SciencesUfaRussia
  2. 2.Département de MathématiquesUniversité Paris SudOrsay CedexFrance

Personalised recommendations