Geodesic
On the exact WKB analysis of microdifferential operators of WKB type
[Sur l'analyse BKW exacte des opérateurs microdifférentiels du type BKW]
Aoki, Takashi1  ; Kawai, Takahiro  ; Koike, Tatsuya ; Takei, Yoshitsugu 
1 Kinki University, School of Science and Engineering, Department of Mathematics, Higashi-Osaka, 577-8502 (Japan), Kyoto University, Institute for Mathematical Sciences, Kyoto, 606-8502 (Japan), Kyoto University, Department of Mathematics, Graduate School of Science, Kyoto, 606-8502 (Japan), Kyoto University, Research Institute for Mathematical Sciences, Kyoto, 606-8502 (Japan)
Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1393-1421

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We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.

Nous introduisons la notion d'opérateur microdifférentiel du type BKW et développons une analyse BKW exacte pour de tels opérateurs en utilisant l'analyse microlocale : nous donnons une méthode récursive pour construire une solution BKW d'un tel opérateur au moyen du calcul symbolique des opérateurs microdifférentiels. Nous étudions leur structure locale au voisinage des points de transition par un théorème de division du type Weierstrass. Nous détaillons l'équation de Berk-Book en appendice.

DOI : 10.5802/aif.2053
Classification : 34M60, 34E20, 34M25, 34M35, 35A27

Aoki, Takashi 1 ; Kawai, Takahiro  ; Koike, Tatsuya  ; Takei, Yoshitsugu 

1 Kinki University, School of Science and Engineering, Department of Mathematics, Higashi-Osaka, 577-8502 (Japan), Kyoto University, Institute for Mathematical Sciences, Kyoto, 606-8502 (Japan), Kyoto University, Department of Mathematics, Graduate School of Science, Kyoto, 606-8502 (Japan), Kyoto University, Research Institute for Mathematical Sciences, Kyoto, 606-8502 (Japan) 
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     title = {On the exact {WKB} analysis of microdifferential operators of {WKB} type},
     journal = {Annales de l'Institut Fourier},
     pages = {1393--1421},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Aoki, Takashi; Kawai, Takahiro; Koike, Tatsuya; Takei, Yoshitsugu. On the exact WKB analysis of microdifferential operators of WKB type. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1393-1421. doi: 10.5802/aif.2053

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