A Landau-Zener Formula for Two-Scaled Wigner Measures

  • Clotilde Fermanian Kammerer
  • Patrick Gerard
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 136)


The semiclassical study of multidimensional crystals leads naturally to the following question: How do Wigner measures propagate through energy level crossings?

In this contribution, we discuss a simple 2 x 2 system which displays such a crossing. For that purpose, we introduce two-scaled Wigner measures, which describe how the usual Wigner transforms are concentrating on trajectories passing through the crossing points. Then we derive explicit formulae for the branching of such measures. These formulae are generalizations of the so-called Landau-Zener formulae. This contribution only contains main statements and some sketch of proofs. Details of proofs will appear in reference [6].


Classical Trajectory Eikonal Equation Semiclassical Limit Cauchy Data Feshbach Resonance 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Alinhac: Branching of singularities for a class of hyperbolic operators. Indiana Univ. Math. J., 27, N° 6 (1978), pp. 1027–1037.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    P. Bechouche and F. Poupaud: Semiclassical limit in a stratified medium, to appear in Monatshefte für Mathematik (2000).Google Scholar
  3. [3]
    A. P. Calderón, R. Vaillancourt: On the boundedness of pseudo-differential operators. J. Math. Soc. Japan, 23, N° 2 (1971), pp. 374–378.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Y. Colin de Verdière: Sur les singularités de Van Hove génériques, Bull. Soc. Math, de France, 119, Mémoire 46 (1991), pp. 99–109.Google Scholar
  5. [5]
    Y. Colin de Verdière, M. Lombardi, and J. Pollet: The microlocal Landau-Zener formula. Ann. Inst. Henri Poincaré, 71, N° 1 (1999), pp. 95–127.MATHGoogle Scholar
  6. [6]
    C. Fermanian Kammerer and P. Gérard: Mesures semi-classiques et croisements de modes, preprint, to appear in Bull. S.M.F.Google Scholar
  7. [7]
    P. Gérard: Mesures semi-classiques et ondes de Bloch, Exposé de l’Ecole Polytechnique, E.D.P., Exposé N° XVI (1991).Google Scholar
  8. [8]
    P. Gérard, P.A. Markowich, N.J. Mauser, and F. Poupaud: Homogenization Limits and Wigner Transforms. Comm. Pure Appl. Math., 50(4) (1997), pp. 323–379.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    G.A. Hagedorn: Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps.Commun. Math. Phys., 136 (1991), pp. 433–449.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    G.A. Hagedorn and A. Joye: Landau-Zener transitions through small electronic eigenvalue gaps in the Born-Oppenheimer approximation. Ann. Inst. Henri Poincaré, 68, N° 1 (1998), pp. 85–134.MathSciNetMATHGoogle Scholar
  11. [11]
    L. Hörmander: The analysis of linear Partial Differential Operators III. Springer-Verlag (1985).Google Scholar
  12. [12]
    A. Joye: Proof of the Landau-Zener formula. Asymptotic Analysis, 9 (1994), pp. 209–258.MathSciNetMATHGoogle Scholar
  13. [13]
    Y. Karpeshina: Perturbation theory for the Schrödinger operator with a periodic potential, Lecture Notes in Mathematics, 1663, Springer, 1997.Google Scholar
  14. [14]
    N. Kaidi and M. Rouleux: Forme normale d’un hamiltonien à deux niveaux près d’un point de branchement (limite semi-classique), C.R. Acad. Sci. Paris Série I Math, 317 (1993), N° 4, pp. 359–364.MathSciNetMATHGoogle Scholar
  15. [15]
    L. Landau: Collected papers of L. Landau, Pergamon Press (1965).Google Scholar
  16. [16]
    P-L. Lions and T. Paul: Sur les mesures de Wigner. Revista Matemática Iberoamericana, 9 (1993), pp. 553–618.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    L. Miller: Propagation d’onde semi-classiques à travers une interface et mesures 2-microlocales. Thèse de VEcole Polytechnique, 1996.Google Scholar
  18. [18]
    P.A. Markowich, N.J. Mauser, and F. Poupaud: A Wigner function approach to semi-classical limits: electrons in a periodic potential, J. Math. Phys., 35 (1994), pp. 1066–1094.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    F. Poupaud and C. Ringhofer: Semi-classical limits in a crystal with exterior potentials and effective mass theorems, Comm. Part. Diff. Eq., 21 (1996), N° 11-12, pp. 1897–1918.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    M. Rouleux: Tunelling effects for h pseudodifferential operators, Feshbach resonances, and the Born-Oppenheimer approximation, Evolution equations, Feshbach resonances, Hodge theory, pp. 131–242, Math. Top., 16, Wiley-VCH, Berlin (1999).Google Scholar
  21. [21]
    C. Wilcox: Theory of Bloch waves, Journal d’Analyse Mathématique, 33 (1978), pp. 146–167.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    C. Zener: Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond., 137 (1932), pp. 696–702.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Clotilde Fermanian Kammerer
    • 1
  • Patrick Gerard
    • 2
  1. 1.MathématiquesUniversité de Cergy-PontoiseCergy-Pontoise cedexFrance
  2. 2.MathématiquesUniversité Paris XIOrsayFrance

Personalised recommendations