no. 17-18
Partial Differential Equations/Mathematical Physics
Transition probability for multiple avoided crossings with a small gap through an exact WKB method and a microlocal approach
Presented by: Jean-Michel Bony
Watanabe, Takuya1; Zerzeri, Maher2
1Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji Higashi, Kusatsu, Shiga, 525-8577, Japan
2LAGA, (UMR CNRS 7539), Institut Galilée, Université Paris Nord 13, 93430 Villetaneuse, France
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 (h↓0) and an interaction parameter (ε↓0). 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” (h↓0) et un paramètre dʼinteraction (ε↓0). 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.

Received:
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

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