The microlocal Landau-Zener formula

 Access to full text

 Full (PDF)

1. [1] J.E. Avron and A. Elgart, An adiabatic theorem without a gap condition, in: Operator Theory: Advances and Applic., Birkhaüser, 1999, pp. 3-12. Zbl0971.81038MR1708784
2. [2] M. Born and V. Fock, Beweis des Adiabatensatzes, Z. Phys.51 (1928) 165-169. Zbl54.0994.03JFM54.0994.03
3. [3] P.J. Braam and J.J. Duistermaat, Normal forms of real symmetric systems with multiplicity, Indag. Math.4 (1993) 407-421. Zbl0802.35176MR1252985
4. [4] M. Carré, A. Zgainsky, M. Gaillard, M. Nouh et M. Lombardi, Détermination des populations relatives des sous-niveaux magnétiques du niveau 4 1 D de HeI excité par impact d'ions lourds, Journal de Physique42 (1981) 235-246. 
5. [5] Y. Colin De Verdière, Limite adiabatique en présence de croisements évités et phases géométriques, 1998, en préparation. 
6. [6] Y. Colin De Verdière et B. Parisse, Equilibre instable en regime semi-classique : I - Concentration microlocale, Commun. PDE 19 (1994) 1535-1563. Zbl0819.35116MR1294470
7. [7] Y. Colin De Verdière et B. Parisse, Equilibre instable en régime semi-classique : II - Conditions de Bohr-Sommerfeld, Ann. Inst. Henri Poincaré (Physique théorique) 61 (1994) 347-367. Zbl0845.35076MR1311072
8. [8] Y. Colin De Verdière et B. Parisse, Singular Boar-Sommerfeld roles, Commun. Math. Phys., to appear. Zbl1157.81310MR1712567
9. [9] Y. Colin De Verdière et J. Vey, Le lemme de Morse isochore, Topology18 (1979) 283-293. Zbl0441.58003MR551010
10. [10] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J.48 (1981) 251-267. Zbl0462.58030MR610185
11. [11] G. Hagedorn, Adiabatic expansions near eigenvalue crossings, Ann. Phys.196 (1989) 278-295. Zbl0875.47002MR1027662
12. [12] G.A. Hagedorn, Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gap, Commun. Math. Phys.136 (1991) 433-449. Zbl0723.35068MR1099690
13. [13] G.A. Hagedorn, Molecular propagation through electron energy level crossings, Memoirs of the AMS536 (1994). Zbl0833.92025MR1234882
14. [14] G.A. Hagedorn and A. Joye, Landau-Zener transitions through small electronic eigenvalues gaps in the Born-Oppenheimer approximation, Ann. Inst. Henri Poincaré (Physique théorique)68 (1998) 85-134. Zbl0915.35090MR1618922
15. [15] A. Joye, Proof of the Landau-Zener formula, Asymptotic Analysis9 (1994) 209- 258. Zbl0814.35109MR1295294
16. [16] A. Joye, Exponential asymptotics in a singular limit for n-level scattering systems, SIAM J. Math. Anal.28 (1997) 669-703. Zbl0991.34071MR1443614
17. [17] S.G. Krein, Linear Differential Equations in Banach Space, Translations of Math. Monographs, Amer. Math. Soc., 1971. MR342804
18. [18] L. Landau, Collected Papers of L. Landau, Pergamon Press, 1965. 
19. [19] P. Martin and G. Nenciu, Semi-classical inelastic S-matrix for one-dimensional N-states systems, Rev. Math. Phys.7 (1995) 193-242. Zbl0835.34098MR1317340
20. [20] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math.32 (1979) 483-519. Zbl0396.58006MR528633
21. [21] A. Messiah, Mecanique Quantique, Dunod, 1969. MR129304
22. [22] J. Pollet, Analyse semi-classique d'un système d'équations de Schrödinger couplées : formule de Landau-Zener. Thèse de l'université de Grenoble1, Octobre 1997. 
23. [23] D. Robert, Autour de l'Approximation Semi-Classique, Birkhäuser, 1987. Zbl0621.35001MR897108
24. [24] H. Rosenthal, Nonadiabatic effects in slow atomic collisions. I. He+ + He, Phys. Rev. A 4 (1971) 1030-1042. 
25. [25] M. Rouleux, Feshbach resonances in the semi-classical limit, Preprint CPT, 1997. 
26. [26] E.C.G. Stueckelberg, Helv. Phys. Acta5 (1932) 369. Zbl0006.09006
27. [27] M. Taylor, Pseudo-differential Operators, Princeton, 1981. 
28. [28] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley, New York, 1985. Zbl0169.10903MR203188
29. [29] C. Zener, Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond.137 (1932) 696-702. Zbl0005.18605

1. Yves Colin de Verdière, The level crossing problem in semi-classical analysis. II. The Hermitian case
2. Clotilde Fermanian Kammerer, Patrick Gérard, Une formule de Landau-Zener pour un croisement générique de codimension 2
3. Gianluca Panati, Herbert Spohn, Stefan Teufel, The time-dependent Born-Oppenheimer approximation
4. Yves Colin de Verdière, The level crossing problem in semi-classical analysis I. The symmetric case
5. Clotilde Fermanian-Kammerer, Patrick Gérard, Mesures semi-classiques et croisement de modes
6. Olivier Lablée, Sur le spectre semi-classique d’un système intégrable de dimension 1 autour d’une singularité hyperbolique
7. Clotilde Fermanian-Kammerer, Patrick Gérard, Mesures semi-classiques et croisement de modes