The microlocal Landau-Zener formula
Annales de l'I.H.P. Physique théorique, Tome 71 (1999) no. 1, p. 95-127
@article{AIHPA_1999__71_1_95_0,
     author = {Colin de Verdi\`ere, Yves and Lombardi, Maurice and Pollet, Jo\"el},
     title = {The microlocal Landau-Zener formula},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {71},
     number = {1},
     year = {1999},
     pages = {95-127},
     zbl = {0986.81027},
     mrnumber = {1704655},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1999__71_1_95_0}
}
Colin de Verdière, Yves; Lombardi, Maurice; Pollet, Joël. The microlocal Landau-Zener formula. Annales de l'I.H.P. Physique théorique, Tome 71 (1999) no. 1, pp. 95-127. https://www.numdam.org/item/AIHPA_1999__71_1_95_0/

[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. | MR 1708784 | Zbl 0971.81038

[2] M. Born and V. Fock, Beweis des Adiabatensatzes, Z. Phys. 51 (1928) 165-169. | JFM 54.0994.03

[3] P.J. Braam and J.J. Duistermaat, Normal forms of real symmetric systems with multiplicity, Indag. Math. 4 (1993) 407-421. | MR 1252985 | Zbl 0802.35176

[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 Physique 42 (1981) 235-246.

[5] Y. Colin De Verdière, Limite adiabatique en présence de croisements évités et phases géométriques, 1998, en préparation.

[6] Y. Colin De Verdière et B. Parisse, Equilibre instable en regime semi-classique : I - Concentration microlocale, Commun. PDE 19 (1994) 1535-1563. | MR 1294470 | Zbl 0819.35116

[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. | Numdam | MR 1311072 | Zbl 0845.35076

[8] Y. Colin De Verdière et B. Parisse, Singular Boar-Sommerfeld roles, Commun. Math. Phys., to appear. | MR 1712567 | Zbl 01379901

[9] Y. Colin De Verdière et J. Vey, Le lemme de Morse isochore, Topology 18 (1979) 283-293. | MR 551010 | Zbl 0441.58003

[10] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J. 48 (1981) 251-267. | MR 610185 | Zbl 0462.58030

[11] G. Hagedorn, Adiabatic expansions near eigenvalue crossings, Ann. Phys. 196 (1989) 278-295. | MR 1027662 | Zbl 0875.47002

[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. | MR 1099690 | Zbl 0723.35068

[13] G.A. Hagedorn, Molecular propagation through electron energy level crossings, Memoirs of the AMS 536 (1994). | MR 1234882 | Zbl 0833.92025

[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. | Numdam | MR 1618922 | Zbl 0915.35090

[15] A. Joye, Proof of the Landau-Zener formula, Asymptotic Analysis 9 (1994) 209- 258. | MR 1295294 | Zbl 0814.35109

[16] A. Joye, Exponential asymptotics in a singular limit for n-level scattering systems, SIAM J. Math. Anal. 28 (1997) 669-703. | MR 1443614 | Zbl 0991.34071

[17] S.G. Krein, Linear Differential Equations in Banach Space, Translations of Math. Monographs, Amer. Math. Soc., 1971. | MR 342804

[18] L. Landau, Collected Papers of L. Landau, Pergamon Press, 1965.

[19] P. Martin and G. Nenciu, Semi-classical inelastic S-matrix for one-dimensional N-states systems, Rev. Math. Phys. 7 (1995) 193-242. | MR 1317340 | Zbl 0835.34098

[20] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math. 32 (1979) 483-519. | MR 528633 | Zbl 0396.58006

[21] A. Messiah, Mecanique Quantique, Dunod, 1969. | MR 129304

[22] J. Pollet, Analyse semi-classique d'un système d'équations de Schrödinger couplées : formule de Landau-Zener. Thèse de l'université de Grenoble 1, Octobre 1997.

[23] D. Robert, Autour de l'Approximation Semi-Classique, Birkhäuser, 1987. | MR 897108 | Zbl 0621.35001

[24] H. Rosenthal, Nonadiabatic effects in slow atomic collisions. I. He+ + He, Phys. Rev. A 4 (1971) 1030-1042.

[25] M. Rouleux, Feshbach resonances in the semi-classical limit, Preprint CPT, 1997.

[26] E.C.G. Stueckelberg, Helv. Phys. Acta 5 (1932) 369. | Zbl 0006.09006

[27] M. Taylor, Pseudo-differential Operators, Princeton, 1981.

[28] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley, New York, 1985. | MR 203188

[29] C. Zener, Non-adiabatic crossing of energy levels, Proc. Roy. Soc. Lond. 137 (1932) 696-702. | Zbl 0005.18605