Advertisement

Annales Henri Poincaré

, Volume 16, Issue 7, pp 1651–1688 | Cite as

Accurate Semiclassical Spectral Asymptotics for a Two-Dimensional Magnetic Schrödinger Operator

  • Bernard Helffer
  • Yuri A. KordyukovEmail author
Article
First Online:

Abstract

We revisit the problem of semiclassical spectral asymptotics for a pure magnetic Schrödinger operator on a two-dimensional Riemannian manifold. We suppose that the minimal value b 0 of the intensity of the magnetic field is strictly positive, and the corresponding minimum is unique and non-degenerate. The purpose is to get the control on the spectrum in an interval \({(hb_0, h(b_0 + \gamma_0)]}\) for some \({\gamma_0 > 0}\) independent of the semiclassical parameter h. The previous papers by Helffer–Mohamed and by Helffer–Kordyukov were only treating the ground-state energy or a finite (independent of h) number of eigenvalues. Note also that N. Raymond and S. Vũ Ngọc have recently developed a different approach of the same problem.

Keywords

Selfadjoint Operator Principal Symbol Semiclassical Analysis Weyl Quantization Weyl Symbol 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Download to read the full article text

References

  1. 1.
    Agmon, S.: Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations: bounds on eigenfunctions of N-body Schrödinger operators. Mathematical Notes, vol. 29. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1982)Google Scholar
  2. 2.
    Bellissard, J.: C * algebras in solid state physics. 2D electrons in a uniform magnetic field. Operator algebras and applications, vol. 2, pp. 49–76, London Math. Soc. Lecture Note Ser., vol. 136. Cambridge Univ. Press, Cambridge (1988)Google Scholar
  3. 3.
    Birman, M. Sh., Solomjak, M.Z.: Spectral theory of selfadjoint operators in Hilbert space. Translated from the 1980 Russian original by S. Khrushchev and V. Peller. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987)Google Scholar
  4. 4.
    Boutet de Monvel L., Grigis A., Helffer B.: Paramétrixes d’opérateurs pseudo-différentiels à caractéristiques multiples. Astérisque 34–35, 93–121 (1976)MathSciNetGoogle Scholar
  5. 5.
    Fournais S., Helffer B.: Accurate eigenvalue asymptotics for Neumann magnetic Laplacians. Ann. Inst. Fourier 56, 1–67 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. Progress in Nonlinear Differential Equations and their Applications, vol. 77. Birkhäuser, Boston (2010)Google Scholar
  7. 7.
    Grushin V.: Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols. Mat. Sb. (N.S.) 88(130), 504–521 (1972)MathSciNetGoogle Scholar
  8. 8.
    Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys, No. 14. AMS, Providence (1977)Google Scholar
  9. 9.
    Helffer B.: Sur l’hypoellipticité des opérateurs à caractéristiques multiples (perte de \({\frac{3}{2}}\) dérivées). Mémoire de la SMF 51–52, 13–61 (1977)MathSciNetGoogle Scholar
  10. 10.
    Helffer, B.: Théorie Spectrale pour des Opérateurs Globalement Elliptiques. Astérisque, vol. 112. Société mathématique de France, Paris (1984)Google Scholar
  11. 11.
    Helffer, B.: Semi-classical Analysis for the Schrödinger Operator and Applications. Lecture Notes in Mathematics, vol. 1336. Springer, Berlin (1988)Google Scholar
  12. 12.
    Helffer, B., Kordyukov, Yu.A.: The periodic magnetic Schrödinger operators: spectral gaps and tunneling effect. Trudy Matematicheskogo Instituta Imeni V.A. Steklova 261, 176–187 (2008); translation in Proceedings of the Steklov Institute of Mathematics, 261, 171–182 (2008)Google Scholar
  13. 13.
    Helffer, B., Kordyukov, Yu.A.: Semiclassical analysis of Schrödinger operators with magnetic wells. Spectral and Scattering Theory for Quantum Magnetic Systems, Contemp. Math. 500, 105–122 (2009)Google Scholar
  14. 14.
    Helffer B., Kordyukov Yu.A.: Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom. J. Funct. Anal. 257, 3043–3081 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Helffer, B., Kordyukov, Yu.A.: Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator: The case of discrete wells. Spectral Theory and Geometric Analysis. Contemp. Math. 535, 55–78 (2011)Google Scholar
  16. 16.
    Helffer B., Kordyukov A.: Semiclassical spectral asymptotics for a two-dimensional magnetic Schrödinger operator II: The case of degenerate wells. Commun. Partial Differ. Equ. 37, 1057–1095 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Helffer, B., Morame, A. (2001) Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185, 604–680 (2001). See erratum in http://www.math.u-psud.fr/~helffer/erratum164II
  19. 19.
    Helffer B., Nourrigat J.: Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs. Birkhäuser, Boston (1985)zbMATHGoogle Scholar
  20. 20.
    Helffer B., Robert D.: Puits de potentiel généralisés et asymptotique semi-classique. Annales de l’IHP (section Physique théorique) 41, 291–331 (1984)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Helffer B., Sjöstrand J.: Multiple wells in the semiclassical limit. I. Commun. Partial Differ. Equ. 9, 337–408 (1984)zbMATHCrossRefGoogle Scholar
  22. 22.
    Helffer B., Sjöstrand J.: Effet tunnel pour l’équation de Schrödinger avec champ magnétique. Ann. Scuola Norm. Sup. Pisa Cl. Sci. Sér. 4 14, 625–657 (1987)zbMATHGoogle Scholar
  23. 23.
    Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Schrödinger operators (Sønderborg, 1988), Lecture Notes in Phys. vol. 345, pp. 118–197. Springer, Berlin (1989)Google Scholar
  24. 24.
    Helffer B., Sjöstrand J.: 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, 113 (1988)Google Scholar
  25. 25.
    Helffer B., Sjöstrand J.: Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum. Mém. Soc. Math. France (N.S.) 39, 1–124 (1989)Google Scholar
  26. 26.
    Hörmander L.: The Weyl calculus of pseudodifferential operators. Commun. Pure Appl. Math. 32, 360–444 (1979)CrossRefGoogle Scholar
  27. 27.
    Ivrii V.: Magnetic Schrödinger operator: classical and quantum dynamics and spectral asymptotics. Mosc. Math. J. 7, 461–479 (2007)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Ivrii, V.: Microlocal Analysis, Sharp Spectral Asymptotics and Applications. Evoluting book available on the home page of V. IvriiGoogle Scholar
  29. 29.
    Karasev M.V.: Magneto-metric Hamiltonians on quantum surfaces in the configuration space. Russ. J. Math. Phys. 14, 57–65 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Matsumoto H.: Semi-classical asymptotics of eigenvalues for Schrödinger operators with magnetic fields. J. Funct. Anal. 129, 168–190 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Matsumoto H., Ueki N.: Spectral analysis of Schrödinger operators with magnetic fields. J. Funct. Anal. 140, 218–255 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Montgomery R.: Hearing the zero locus of a magnetic field. Commun. Math. Phys. 168, 651–675 (1995)zbMATHADSCrossRefGoogle Scholar
  33. 33.
    Raymond, N.: Little Magnetic Book (2014). Preprint arXiv:1405.7912
  34. 34.
    Raymond, N., Vũ Ngọc, S.: Geometry and spectrum in 2D magnetic wells (2013). Preprint arXiv:1306.5054; to appear in Ann. Inst. Fourier (2014)
  35. 35.
    Robert, D.: Autour de l’Approximation Semiclassique. Progress in Mathematics, vol. 68. Birkhäuser, Boston (1987)Google Scholar
  36. 36.
    Shubin M.: Pseudodifferential Operators and Spectral Theory. Springer, Berlin (1987)zbMATHCrossRefGoogle Scholar
  37. 37.
    Sjöstrand J.: Parametrices for pseudo-differential operators with multiple characteristics. Ark. för Mat. 12, 85–130 (1974)zbMATHCrossRefGoogle Scholar
  38. 38.
    Sjöstrand J., Zworski M.: Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier 57, 1–33 (2007)CrossRefGoogle Scholar
  39. 39.
    Zworski M.: Semiclassical Analysis. AMS, Providence (2012)zbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Département de Mathématiques, Bâtiment 425Univ. Paris-Sud et CNRSOrsay CédexFrance
  2. 2.Institute of MathematicsRussian Academy of SciencesUfaRussia

Personalised recommendations