no. 5
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.

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.

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.

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) 
@article{AIF_2004__54_5_1393_0,
     author = {Aoki, Takashi and Kawai, Takahiro and Koike, Tatsuya and Takei, Yoshitsugu},
     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},
     volume = {54},
     number = {5},
     year = {2004},
     doi = {10.5802/aif.2053},
     mrnumber = {2127852},
     zbl = {1079.34070},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2053/}
}
TY  - JOUR
AU  - Aoki, Takashi
AU  - Kawai, Takahiro
AU  - Koike, Tatsuya
AU  - Takei, Yoshitsugu
TI  - On the exact WKB analysis of microdifferential operators of WKB type
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 1393
EP  - 1421
VL  - 54
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2053/
DO  - 10.5802/aif.2053
LA  - en
ID  - AIF_2004__54_5_1393_0
ER  - 
%0 Journal Article
%A Aoki, Takashi
%A Kawai, Takahiro
%A Koike, Tatsuya
%A Takei, Yoshitsugu
%T On the exact WKB analysis of microdifferential operators of WKB type
%J Annales de l'Institut Fourier
%D 2004
%P 1393-1421
%V 54
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2053/
%R 10.5802/aif.2053
%G en
%F AIF_2004__54_5_1393_0
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. http://archive.numdam.org/articles/10.5802/aif.2053/

[A] T. Aoki, Quantized contact transformations and pseudodifferential operators of infinite order, Publ. RIMS, Kyoto Univ. 26 (1990) p. 505-519 | MR | Zbl

[AKKT1] T. Aoki, T. Kawai, T. Koike & Y. Takei, On the exact WKB analysis of operators admitting infinitely many phases, Adv. Math. 181 (2004) p. 165-189 | MR | Zbl

[AKKT2] T. Aoki, T. Kawai, T. Koike & Y. Takei, On global aspects of exact WKB analysis of operators admitting infinitely many phases, RIMS Preprint 1392, 2003 | MR | Zbl

[AKKT3] T. Aoki, T. Kawai, T. Koike & Y. Takei, On the exact WKB analysis of microdifferential operators (in Japanese), RIMS Kôkyûroku 1316 (2003)

[AY] T. Aoki & J. Yoshida, Microlocal reduction of ordinary differential operators with a large parameter, Publ. RIMS, Kyoto Univ. 29 (1993) p. 959-975 | MR | Zbl

[BB] H.L. Berk & D.L. Book, Plasma wave regeneration in inhomogeneous media, Phys. Fluids 12 (1969) p. 649-661 | Zbl

[BNR] H.L. Berk, W.M. Nevins & K.V. Roberts, New Stokes' line in WKB theory, J. Math. Phys. 23 (1982) p. 988-1002 | MR | Zbl

[BRS] H.L. Berk, M.N. Rosenbluth & R.N. Sudan, Plasma wave propagation in hot inhomogeneous media, Phys. Fluids 9 (1966) p. 1606-1608

[BK] L. Boutet De Monvel & P. Krée, Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier 17 (1967) p. 295-323 | Numdam | MR | Zbl

[DDP] E. Delabaere, H. Dillinger & F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier 43 (1993) p. 163-199 | Numdam | MR | Zbl

[KT] T. Kawai & Y. Takei, Algebraic Analysis of Singular Perturbations (in Japanese), 1998

[KoT] T. Koike & Y. Takei, On the zero-set of some entire function of two complex variables arising from a problem in physics, in preparation

[L] L. Landau, On the vibrations of the electronic plasma, J. Phys. USSR 10 (1946) p. 25-34 | MR | Zbl

[M] B. Malgrange, L'involutivité des caractéristiques des systèmes différentiels et microdifférentiels, Sém. Bourbaki 1977/1978 522, | Numdam | Zbl

[Mar] A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, Springer, 2002 | MR | Zbl

[P] F. Pham, Multiple turning points in exact WKB analysis (variations on a theme of Stokes), Kyoto Univ. Press, 2000, p. 71-85 | Zbl

[S] H. J. Silverstone, JWKB connection-formula problem revisited via Borel summation, Phys. Rev. Lett. 55 (1985) p. 2523-2526 | MR

[Sj1] J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982) | Numdam | MR | Zbl

[Sj2] J. Sjöstrand, Projecteurs adiabatiques du point de vue pseudo différentiel, C. R. Acad. Sci. Paris, Sér. I 317 (1993) p. 217-220 | MR | Zbl

[V] A. Voros, The return of the quartic oscillator -- The complex WKB method, Ann. Inst. Henri Poincaré 39 (1983) p. 211-338 | Numdam | MR | Zbl

[<L>L</L>] L. Landau, On the vibrations of the electronic plasma, Pergamon Press, 1965, p. 445-460

Cité par Sources :