On the exact WKB analysis of microdifferential operators of WKB type

Takashi Aoki[1]; Takahiro Kawai; Tatsuya Koike; Yoshitsugu Takei

  • [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 (2004)

  • Volume: 54, Issue: 5, page 1393-1421
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Aoki, Takashi, et al. "On the exact WKB analysis of microdifferential operators of WKB type." Annales de l’institut Fourier 54.5 (2004): 1393-1421. <http://eudml.org/doc/116146>.

@article{Aoki2004,
abstract = {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.},
affiliation = {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)},
author = {Aoki, Takashi, Kawai, Takahiro, Koike, Tatsuya, Takei, Yoshitsugu},
journal = {Annales de l’institut Fourier},
keywords = {exact WKB analysis; microdifferential operators of WKB type; turning points; Weierstrass-type division theorem for microdifferential operators; Berk-Book equation},
language = {eng},
number = {5},
pages = {1393-1421},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the exact WKB analysis of microdifferential operators of WKB type},
url = {http://eudml.org/doc/116146},
volume = {54},
year = {2004},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1393
EP - 1421
AB - 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.
LA - eng
KW - exact WKB analysis; microdifferential operators of WKB type; turning points; Weierstrass-type division theorem for microdifferential operators; Berk-Book equation
UR - http://eudml.org/doc/116146
ER -

References

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  1. T. Aoki, Quantized contact transformations and pseudodifferential operators of infinite order, Publ. RIMS, Kyoto Univ. 26 (1990), 505-519 Zbl0719.58037MR1068863
  2. T. Aoki, T. Kawai, T. Koike, Y. Takei, On the exact WKB analysis of operators admitting infinitely many phases, Adv. Math. 181 (2004), 165-189 Zbl1056.34103MR2020659
  3. T. Aoki, T. Kawai, T. Koike, Y. Takei, On global aspects of exact WKB analysis of operators admitting infinitely many phases, (2003) Zbl1087.34064MR2130824
  4. T. Aoki, T. Kawai, T. Koike, Y. Takei, On the exact WKB analysis of microdifferential operators (in Japanese), RIMS Kôkyûroku 1316 (2003) Zbl1079.34070
  5. T. Aoki, J. Yoshida, Microlocal reduction of ordinary differential operators with a large parameter, Publ. RIMS, Kyoto Univ. 29 (1993), 959-975 Zbl0807.34071MR1256439
  6. H.L. Berk, D.L. Book, Plasma wave regeneration in inhomogeneous media, Phys. Fluids 12 (1969), 649-661 Zbl0179.59101
  7. H.L. Berk, W.M. Nevins, K.V. Roberts, New Stokes' line in WKB theory, J. Math. Phys. 23 (1982), 988-1002 Zbl0488.34050MR659998
  8. H.L. Berk, M.N. Rosenbluth, R.N. Sudan, Plasma wave propagation in hot inhomogeneous media, Phys. Fluids 9 (1966), 1606-1608 
  9. L. Boutet de Monvel, P. Krée, Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier 17 (1967), 295-323 Zbl0195.14403MR226170
  10. E. Delabaere, H. Dillinger, F. Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Ann. Inst. Fourier 43 (1993), 163-199 Zbl0766.34032MR1209700
  11. T. Kawai, Y. Takei, Algebraic Analysis of Singular Perturbations (in Japanese), (1998), Iwanami, Tokyo Zbl0938.34532
  12. T. Koike, Y. Takei, On the zero-set of some entire function of two complex variables arising from a problem in physics 
  13. L. Landau, On the vibrations of the electronic plasma, J. Phys. USSR 10 (1946), 25-34 Zbl0063.03439MR23765
  14. B. Malgrange, L'involutivité des caractéristiques des systèmes différentiels et microdifférentiels, 522 Zbl0423.46033
  15. A. Martinez, An Introduction to Semiclassical and Microlocal Analysis, (2002), Springer, New York Zbl0994.35003MR1872698
  16. F. Pham, Multiple turning points in exact WKB analysis (variations on a theme of Stokes), Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear (2000), 71-85, Kyoto Univ. Press Zbl1017.34091
  17. H. J. Silverstone, JWKB connection-formula problem revisited via Borel summation, Phys. Rev. Lett. 55 (1985), 2523-2526 MR819680
  18. J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982) Zbl0524.35007MR699623
  19. J. Sjöstrand, Projecteurs adiabatiques du point de vue pseudo différentiel, C. R. Acad. Sci. Paris, Sér. I 317 (1993), 217-220 Zbl0783.35087MR1231425
  20. A. Voros, The return of the quartic oscillator -- The complex WKB method, Ann. Inst. Henri Poincaré 39 (1983), 211-338 Zbl0526.34046MR729194
  21. L. Landau, On the vibrations of the electronic plasma, Collected papers of L.D. Landau (1965), 445-460, Pergamon Press Zbl0063.03439

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