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

Takuya Watanabe; Maher Zerzeri

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

Takuya Watanabe ¹ ; Maher Zerzeri 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, Volume 350 (2012) no. 17-18, pp. 841-844.

Abstract (VO)
Résumé

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.

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 :

Takuya Watanabe ¹ ; Maher Zerzeri 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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     author = {Takuya Watanabe and Maher Zerzeri},
     title = {Transition probability for multiple avoided crossings with a small gap through an exact {WKB} method and a microlocal approach},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {841--844},
     publisher = {Elsevier},
     volume = {350},
     number = {17-18},
     year = {2012},
     doi = {10.1016/j.crma.2012.10.005},
     language = {en},
}

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Takuya Watanabe; Maher Zerzeri. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.005/

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