Semiclassical analysis of generic codimension 3 crossings

International Mathematics Research Notices

Volumn , Issue 45, 2004, Pages 2391-2435

  Kammerer, Clotilde Fermanian ^a  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

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[No Author keywords available]

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EID: 11144342763     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: 10.1155/s1073792804133175     Document Type: Article

References (37)

1. P. Bechouche, N. J. Mauser, and F. Poupaud, Semiclassical limit for the Schrödinger-Poisson equation in a crystal, Comm. Pure Appl. Math. 54 (2001), no. 7, 851-890.
2. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and Its Applications, vol. 5, North-Holland Publishing, Amsterdam, 1978.
3. P. J. Braam and J. J. Duistermaat, Normal forms of real symmetric systems with multiplicity, Indag. Math. (N.S.) 4 (1993), no. 4, 407-421.
4. A.-P. Calderón and R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971), no. 2, 374-378.
5. Y. Colin de Verdière, Sur les singularitès de van Hove génériques [On generic van Hove singularities], Mém. Soc. Math. Fr. (N.S.) 119 (1991), no. 46, 99-110 (French).
6. _, The level crossing problem in semi-classical analysis I. The symmetric class, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 4, 1023-1054.
7. _, The level crossing problem in semi-classical analysis H. The Hermitian case, to appear in Ann. Inst. Fourier (Grenoble).
8. C. Fermanian Kammerer, A non commutative Landau-Zener formula, to appear in Math. Nach.
9. _, Semi-classical analysis of a Dirac equation without adaibatic decoupling, to appear in Monat. Für Math.
10. _, Wigner measures and molecular propagation through generic energy level crossings, to appear in Rev. Math. Phys.
11. C. Fermanian Kammerer and P. Gérard, Une formule de Landau-Zener pour un croisement générique de codimension 2, Séminaire sur les Équations aux Dérivées Partielles, 2001-2002, Exp. no. 21, École Polytechnique, Palaiseau.
12. _, Mesures semi-classiques et croisement de modes [Semiclassical measures and eigen-value crossings], Bull. Soc. Math. France 130 (2002), no. 1, 123-168 (French).
13. _, A Landau-Zener formula for non-degenerated involutive codimension 3 crossings, Ann. Henri Poincaré 4 (2003), no. 3, 513-552.
14. C. Fermanian Kammerer and C. Lasser, Wigner measures and codimension two crossings, J. Math. Phys. 44 (2003), no. 2, 507-527.
15. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, International Series in Pure and Applied Physics, McGraw-Hill Publishing, Maidenhead, Berkshire, 1965.
16. P. Gérard, Mesures semi-classiques et ondes de Bloch [Semiclassical measures and Bloch waves], Séminaire sur les Équations aux Dérivées Partielles, 1990-1991, Exp. no. 16, École Polytechnique, Palaiseau.
17. _, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), no. 11, 1761-1794.
18. P. Gérard and É. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (1993), no. 2, 559-607.
19. P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Erratum: "Homogenization limits and Wigner transforms", [Comm. Pure Appl. Math. 50 (1997), no. 4, 323-379], Comm. Pure Appl. Math. 53 (2000), no. 2, 280-281.
20. P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Erratum: "Homogenization limits and Wigner transforms", [Comm. Pure Appl. Math. 50 (1997), no. 4, 323-379], Comm. Pure Appl. Math. 53 (2000), no. 2, 280-281.
21. G. A. Hagedorn, Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps, Comm. Math. Phys. 136 (1991), no. 3, 433-449.
22. _, Molecular propagation through electron energy level crossings, Mem. Amer. Math. Soc. 111 (1994), no. 536, vi+130.
23. G. A. Hagedorn and A. Joye, Landau-Zener transitions through small electronic eigenvalue gaps in the Born-Oppenheimer approximation, Ann. Inst. H. Poincaré Phys. Théor. 68 (1998), no. 1, 85-134.
24. _, Molecular propagation through small avoided crossings of electron energy levels, Rev. Math. Phys. 11 (1999), no. 1, 41-101.
25. L. Hönnander, The Analysis of Linear Partial Differential Operators. III, Grundlehren der mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin, 1985.
26. A. Joye, Proof of the Landau-Zener formula, Asymptot. Anal. 9 (1994), no. 3, 209-258.
27. L. Landau, Collected Papers of L. Landau, Pergamon Press, Oxford, 1965.
28. P.-L. Lions and T. Paul, Sur les mesures de Wigner [On Wigner measures], Rev. Mat. Iberoamericana 9 (1993), no. 3, 553-618 (French).
29. 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), no. 3, 1066-1094.
30. L. Miller, Propagation d'ondes semiclassiques à travers une interface et mesures 2-microlocales, Ph.D. Thesis, École Polytechnique, Paris, 1996.
31. E. Nelson, Topics in Dynamics. I: Flows, Mathematical Notes, Princeton University Press, New Jersey, 1969.
32. F. Poupaud and C. Ringhofer, Semi-classical limits in a crystal with exterior potentials and effective mass theorems, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1897-1918.
33. D. Robert, Autour de l'Approximation Semi-classique [On Semiclassical Approximation], Progress in Mathematics, vol. 68, Birkhäuser Boston, Massachusetts, 1987.
34. E. R. Tracy and A. N. Kaufman, Ray helicity: geometric invariant for multidimensional resonant wave conversion, Phys. Rev. Lett. 91 (2003), no. 13, 130402.
35. E. R. Tracy, A. N. Kaufman, and A. J. Brizard, Ray-based methods in multi-dimensional linear wave conversion, Phys. Plasmas 10 (2003), 2147-2154.
36. A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math. 6 (1971), 329-346.
37. C. Zener, Nan-adiabatic crossing of energy levels, Proc. Roy. Soc. London Ser. A 137 (1932), 696-702.