Partial Differential Equations/Mathematical Physics
Transition probability for multiple avoided crossings with a small gap through an exact WKB method and a microlocal approach
Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 841-844.

In this Note, we study the adiabatic transition probability for a two-level system in the case of a finite number of avoided crossings. More precisely, we investigate a global change of bases of a first order differential system with respect to a semiclassical “adiabatic” parameter (h0) and an interaction parameter (ε0). We obtain its asymptotic behaviors by means of an exact WKB method and a microlocal analysis according to the interrelation of the two parameters.

Dans cette Note, nous intéressons à la probabilité de transition adiabatique dʼun système à deux niveaux dans le cas dʼun nombre fini de croisements évités. Plus précisément, nous étudions un changement global des bases dʼun système différentiel du premier ordre par rapport à un paramètre semiclassique “adiabatique” (h0) et un paramètre dʼinteraction (ε0). Nous obtenons les différents comportements asymptotiques au moyen dʼune méthode BKW exacte et une analyse microlocale en fonction de la corrélation entre les deux paramètres.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2012.10.005
Watanabe, Takuya 1; Zerzeri, Maher 2

1 Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji Higashi, Kusatsu, Shiga, 525-8577, Japan
2 LAGA, (UMR CNRS 7539), Institut Galilée, Université Paris Nord 13, 93430 Villetaneuse, France
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Watanabe, Takuya; Zerzeri, Maher. Transition probability for multiple avoided crossings with a small gap through an exact WKB method and a microlocal approach. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 841-844. doi : 10.1016/j.crma.2012.10.005. http://www.numdam.org/articles/10.1016/j.crma.2012.10.005/

[1] Colin de Verdière, Y.; Lombardi, M.; Pollet, J. The microlocal Landau–Zener formula, Ann. Inst. Henri Poincaré, Volume 71 (1999), pp. 95-127

[2] Fujiié, S. Semiclassical representation of the scattering matrix by a Feynman integral, Comm. Math. Phys., Volume 198 (1998) no. 2, pp. 407-425

[3] Fujiié, S.; Lasser, C.; Nédélec, L. Semiclassical resonances for a two-level Schrödinger operator with a conical intersection, Asymptot. Anal., Volume 65 (2009) no. 1–2, pp. 17-58

[4] Gérard, C.; Grigis, A. Precise estimates of tunneling and eigenvalues near a potential barrier, J. Differential Equations, Volume 42 (1988), pp. 149-177

[5] Hagedorn, G.-A. Proof of the Landau–Zener formula in an adiabatic limit with small eigenvalue gaps, Comm. Math. Phys., Volume 136 (1991) no. 4, pp. 33-49

[6] G.-A. Hagedorn, A. Joye, Recent results on non-adiabatic transitions in quantum mechanics, in: Proceedings of the 2005 UAB International Conference on Differential Equations and Mathematical Physics, Birmingham, Alabama, March 29–April 2, 2005.

[7] Helffer, B.; Sjöstrand, J. Semiclassical analysis for Harperʼs equation, III. Cantor structure of the spectrum, Mém. Soc. Math. France, Volume 39 (1989), pp. 1-124

[8] Joye, A. Proof of the Landau–Zener formula, Asymptot. Anal., Volume 9 (1994), pp. 209-258

[9] Joye, A.; Mileti, G.; Pfister, C.-E. Interferences in adiabatic transition probabilities mediated by Stokes lines, Phys. Rev. A, Volume 44 (1991), pp. 4280-4295

[10] Martinez, A. Precise exponential estimates in adiabatic theory, J. Math. Phys., Volume 35 (1994), pp. 3889-3915

[11] Rousse, V. Landau–Zener transitions for eigenvalue avoided crossings, Asymptot. Anal., Volume 37 (2004) no. 3–4, pp. 293-328

[12] Watanabe, T. Adiabatic transition probability for a tangential crossing, Hiroshima Math. J., Volume 36 (2006) no. 3, pp. 443-468

[13] Zener, C. Non-adiabatic crossing of energy levels, Proc. R. Soc. London A, Volume 137 (1932), pp. 696-702

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