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Two-scale Wigner Measures and the Landau—Zener Formulas

  • C. Fermanian Kammerer
  • P. Gérard
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

We consider the system of evolution equations
$$i\varepsilon \frac{{\partial {{\psi }^{\varepsilon }}}}{{{{\partial }_{t}}}} = o{{p}_{\varepsilon }}(H){{\psi }^{\varepsilon }},$$
where ε is a small parameter, ψεε(t,x) a vector-valued bounded family in L 2(ℝ d ), H = H(t,x,ξ) a matrix-valied Hamiltonian.The variable x denotes the position variable and ξ the momentum.We use Weyl quantization:
$$o{{p}_{\varepsilon }}(a) = \int_{{{{\mathbb{R}}^{d}} \times {{\mathbb{R}}^{d}}}} {{{e}^{{i(x - y) \cdot \xi }}}a} \left( {\frac{{x + y}}{2},\varepsilon \xi } \right)f(y)\frac{{dyd\xi }}{{{{{(2\pi )}}^{d}}}}.$$

Keywords

Electron Energy Level Molecular Propagation Hamiltonian Flow Weyl Quantization Critical Trajectory 
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.

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References

  1. 1.
    P.J. Braam, J.J. Duistermaat, Normal forms of real symmetric systems with multiplicity Indag. Mathem. N.S., 4(4) (1993), 407–421.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Y. Colin de Verdière, The level crossing problem in semi-classical analysis I. The symmetric case Ann. Inst. Fourier 53(2003), 1023–1054.MATHCrossRefGoogle Scholar
  3. 3.
    Y. Colin de Verdière, The level crossing problem in semi-classical analysis II. The hermitian case Preprint de l’Institut Fourier 2003.Google Scholar
  4. 4.
    C. Fermanian Kammerer, A non-commutative Landau-Zener formula. Prépublication de l’Université de Cergy-Pontoise2002, to appear in Math. Nach. Google Scholar
  5. 5.
    C. Fermanian Kammerer, Wigner measures and Molecular Propagation through Generic Energy Level Crossings Prépublication de l’Université de Cergy-Pontoise, October 2002.Google Scholar
  6. 6.
    C. Fermanian Kammerer, Semi-classical analysis of a Dirac equation without adaibatic decoupling, Prépublication de l’Université de Cergy-Pontoise 2003; to appear in Monat. Für Math. Google Scholar
  7. 7.
    C. Fermanian Kammerer, P. Gérard, Mesures semi-classiques et croisements de modes Bull. Soc. Math. France 130 No;1, (2002), 123–168.Google Scholar
  8. 8.
    C. Fermanian Kammerer, P. Gérard, Une formule de Landau-Zener pour un croisement générique de codimension 2, Séminaire E.D.P. 2001–2002, Exposé No;21, Ecole Polytechnique. http://math.polytechnique.fr/seminaires/seminaires-edpGoogle Scholar
  9. 9.
    C. Fermanian Kammerer, P. Gérard, A Landau-Zener formula for non-degenerated involutive codimension 3 crossing Prépublication de l’Université de Cergy-Pontoise Sept. 2002; Annales Henri Poincaré 4(2003), 513–552.MATHCrossRefGoogle Scholar
  10. 10.
    C. Fermanian Kammerer, C. Lasser, Wigner measures, codimension two crossings, Jour. Math. Phys. 44 2 (2003), 507–527.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    P. Gérard, P. A. Markowich, N. J. Mauser, F. Poupaud, Homogenization Limits and Wigner Transforms, Comm. Pure Appl. Math., 50 (1997), 4, 323–379 and 53 (2000), 280–281.CrossRefGoogle Scholar
  12. 12.
    G.A. Hagedorn, Proof of the Landau—Zener formula in an adiabatic limit with small eigenvalue gaps Commun. Math. Phys 136(1991), 433–449.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    G.A. Hagedorn, Molecular Propagation through Electron Energy Level Crossings Memoirs of the A. M. S. 111N° 536, (1994).Google Scholar
  14. 14.
    G.A. Hagedorn, A. Joye, Landau—Zener transitions through small electronic eigenvalue gaps in the Born—Oppenheimer approximation Ann. Inst. Henri Poincaré, Physique Théorique, 4 No1, (1998), 85–134.Google Scholar
  15. 15.
    G. A. Hagedorn, A. Joye, Molecular propagation through small avoided crossings of electron energy levels Rev. Math. Phys. 11 N° 1, (1999), 41–101.MathSciNetCrossRefGoogle Scholar
  16. 16.
    L. Landau: Collected Papers of L. Landau Pergamon Press, 1965.Google Scholar
  17. 17.
    P-L. Lions, T. Paul: Sur les mesures de Wigner Revista Matemática Iberoamericana 9 (1993), 553–618.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    C. Zener: Non-adiabatic crossing of energy levels Proc. Roy. Soc. Lond. 137 (1932),696–702.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • C. Fermanian Kammerer
    • 1
  • P. Gérard
    • 2
  1. 1.Université de Cergy-Pontoise 2 av. Adolphe ChauvinCergy Pontoise CedexFrance
  2. 2.Laboratoire de Mathématiques d’OrsayUniversité Paris-Sud - Bât 425Orsay CedexFrance

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