Transition probability for multiple avoided crossings with a small gap through an exact WKB method and a microlocal approach

Watanabe, Takuya; Zerzeri, Maher

doi:10.1016/j.crma.2012.10.005

Partial Differential Equations/Mathematical Physics

Transition probability for multiple avoided crossings with a small gap through an exact WKB method and a microlocal approach
[Probabilité de transition pour de multiples croisements évités avec un petit écart via la méthode BKW exacte et lʼapproche microlocale]

Présenté par : Jean-Michel Bony

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

Comptes Rendus. Mathématique, Tome 350 (2012) no. 17-18, pp. 841-844.

Résumé
Abstract

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.

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.

Reçu le : 2012-06-16
Accepté le : 2012-10-03
Publié le : 2012-09-01

DOI : 10.1016/j.crma.2012.10.005

Affiliations des auteurs :

Watanabe, Takuya ¹ ; Zerzeri, Maher ²

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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, Tome 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/

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