Non-trapping condition for semiclassical Schrödinger operators with matrix-valued potentials

Mathematical Physics Electronic Journal

Volumn 11, Issue , 2005, Pages

  Jecko, Thierry ^a  

^a UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Coherent states; Egorov's theorem; Eigenvalues crossing; Mourre theory; Non trapping condition; Schr dinger matrix operators; Semiclassical measure; Semiclassical resolvent estimates

Indexed keywords

EID: 17444428119     PISSN: 10866655     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (33)

1. [Ba] A. Balazard-Konlein: Calcul fonctionnel pour des opérateurs h-admissibles à symbole opérateur et applications. PhD Thesis, Université de Nantes, 1985, and also C.R. Acad. Sci. Paris, Série I 301, 903-906, 1985.
2. [Ba] A. Balazard-Konlein: Calcul fonctionnel pour des opérateurs h-admissibles à symbole opérateur et applications. PhD Thesis, Université de Nantes, 1985, and also C.R. Acad. Sci. Paris, Série I 301, 903-906, 1985.
3. [BG] J. Boite, R. Glaser: A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators, preprint, arXiv:math-ph/0204018, 2002.
4. [BN] R. Brummelhuis, J. Nourrigat: Scattering amplitude for Dirac operators. Comm. PDE 24 (1 & 2), 377-394 (1999).
5. [B] N. Burq: Semiclassical estimates for the resolvent in non trapping geometries. Int. Math. Res. Notices 2002, 5, 221-241.
6. [CLP] Y. Colin de Verdière, M. Lombardi, J. Pollet: The microlocal Landau-Zener formula. Ann. Inst. Henri Poincaré, Phys. Théor. 71, 1, 95-127 (1999).
7. [CFKS] H. L. Cycon, R. Froese, W. Kirsch, B. Simon: Schrödinger operators with application to quantum mechanics and global geometry. Springer-Verlag 1987.
8. [DG] J. Dereziński, C. Gérard: Scattering theory of classical and quantum N-particle systems. Springer-Verlag 1997.
9. [FG] C. Fermanian-Kammerer, P. Gérard: Mesures semi-classiques et croisement de modes. Bull. Soc. math. France 130 (1), 2002, p. 123-168.
10. [FH] R. Froese, L Herbst: Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators. Comm. Math. Phys. 87, 3, p. 429-447 (1982/1983).
11. [GM] C. Gérard, A. Martinez: Principe d'absorption limite pour des opérateurs de Schrödinger à longue portée. C.R. Acad. Sci. Paris, Série I 306, 121-123, 1988.
12. [GL] P. Gérard, E. Leichtnam: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71, 2, 1993.
13. [H] G. A. Hagedorn: Molecular propagation through electron energy level crossings. Memoirs AMS 536, vol. 111, 1994.
14. [J1] Th. Jecko: Estimations de la résolvante pour une molécule diatomique dans l'approximation de Born-Oppenheimer. Comm. Math. Phys. 195, 3, 585-612, 1998.
15. [J2] Th. Jecko: Classical limit of elastic scattering operator of a diatomic molecule in the Born-Oppenheimer approximation. Ann. Inst. Henri Poincaré, Physique théorique, 69, 1, 1998, p. 83-131.
16. [J3] Th. Jecko: Approximation de Born-Oppenheimer de sections efficaces totales diatomiques. Asympt. Anal. 24 (2000), p. 1-35.
17. [J4] Th. Jecko: Semiclassical resolvent estimates for Schrödinger matrix operators with eigenvalues crossing., Math. Nachr. 257, 36-54 (2003).
18. [J5] Th. Jecko: From classical to semiclassical non-trapping behaviour. C. R. Acad. Sci. Paris, Ser. I 338 (2004), p. 545-548.
19. [Ka] U. Karlsson: Semi-classical approximations of quantum mechanical problems. PhD thesis, KTH, Stockholm 2002.
20. [K] T. Kato: Pertubation theory for linear operators. Springer-Verlag 1995.
21. [KMW1] M. Klein, A. Martinez, X. P. Wang: On the Born-Oppenheimer approximation of wave operators in molecular Scattering. Comm. Math. Phys 152, 73-95 (1993).
22. [KMW2] M. Klein, A. Martinez, X. P. Wang: On the Born-Oppenheimer approximation of diatomic wave operators. II. Singular potentials. J. Math. Phys. 38 (3), March 1997.
23. [Ma] A. Martinez: Resonance free domains for non globally analytic potentials. Ann. H. Poincaré 3 (2002), no. 4, p. 739-756.
24. [Mo] E. Mourre: Absence of singular continuous spectrum for certain self-adjoint operators. Comm. Math. Phys. 78, 391-408, 1981.
25. [Né] L. Nédélec: Resonances for matrix Schrödinger operators. Duke Math. J. 106, 2, 209-236 (2001).
26. [N] F. Nier: A semiclassical picture of quantum scattering. Ann. Sci. École Norm. Sup. (4) 29 (1996), 2, 149-183.
27. [RS2] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Tome II: Fourier Analysis, Self-adjointness. Academic Press 1979.
28. [RS4] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Tome IV: Analysis of operators. Academic Press 1979.
29. [R] D. Robert: Autour de l'approximation semi-classique. Birkhäuser 1987.
30. [RT] D. Robert, H. Tamura: Semiclassical estimates for resolvents and asymptotics for total cross-section. Ann. IHP 46, 1987, pp. 415-442.
31. [S] V. Sordoni: Reduction scheme for semiclassical operator-valued Schrödinger type equation and application to scattering. Comm. PDE, 28, 7 & 8, pp. 1221-1236, 2003.
32. [VZ] A. Vasy, M. Zworski: Semiclassical estimates in asymptotically euclidean scattering. Comm. Math. Phys. 212, 1, pp. 205-217 (2000).
33. [W] X. P. Wang: Semiclassical resolvent estimates for N-body Schrödinger operators. J. Funct. Anal. 97, 466-483 (1991).