Virtual turning points — A gift of microlocal analysis to the exact WKB analysis

  • Takashi Aoki
  • Naofumi Honda
  • Takahiro Kawai
  • Tatsuya Koike
  • Yukihiro Nishikawa
  • Shunsuke Sasaki
  • Akira Shudo
  • Yoshitsugu Takei


Several aspects of the notion of virtual turning points are discussed; its background, its relevance to the bifurcation phenomena of a Stokes curve, its importance in the analysis of the Noumi-Yamada system (a particular higher order Painlevé equation) and a concrete recipe for locating them. Examples given here make it manifest that virtual turning points are indispensable in WKB analysis of higher order linear ordinary differential equations with a large parameter.


Turning Point Microlocal Analysis Instanton Expansion Stokes Phenomenon High Order Linear 
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  1. [AKKSST]
    T. Aoki, T. Kawai, T. Koike, S. Sasaki, S. Shudo and Y. Takei: A background story and some know-how of virtual turning points, RIMS Koukyuuroku, (ISSN 1880-2818), No.1424, 2005, pp.53–63.Google Scholar
  2. [AKKT]
    T. Aoki, T. Kawai, T. Koike and Y. Takei: On global aspects of exact WKB analysis of operators admitting infinitely many phases. Contemporary Math., No.373, 2005, pp. 11–47.MathSciNetGoogle Scholar
  3. [AKSST]
    T. Aoki, T. Kawai, S. Sasaki, A. Shudo and Y. Takei: Virtual turning points and bifurcation of Stokes curves, J. Phys. A: Math. Gen., 38 (2005), 3317–3336.MATHCrossRefMathSciNetGoogle Scholar
  4. [AKT1]
    T. Aoki, T. Kawai and Y. Takei: New turning points in the exact WKB analysis for higher-order ordinary differential equations, Analyse Algébrique des Perturbations Singulières. I, Hermann, 1994, pp. 69–84.MathSciNetGoogle Scholar
  5. [AKT2]
    _____: On the exact WKB analysis for the third order ordinary differential equations with a large parameter, Asian J. Math., 2 (1998), 625–640.MATHMathSciNetGoogle Scholar
  6. [AKT3]
    _____: On the exact steepest descent method: A new method for the description of Stokes curves, J. Math. Phys., 42 (2001), 3691–3713.MATHCrossRefMathSciNetGoogle Scholar
  7. [AKT4]
    _____: The exact steepest descent method — A new steepest descent method based on the exact WKB analysis, Advanced Studies in Pure Math., No.42, 2004, pp. 45–61.MathSciNetGoogle Scholar
  8. [AKT5]
    _____: Exact WKB analysis of non-adiabatic transition probabilities for three levels, J. Phys. A: Math. Gen., 35 (2002), 2401–2430.MATHCrossRefMathSciNetGoogle Scholar
  9. [BW]
    C. M. Bender and T. T. Wu: Anharmonic oscillator, Phys. Rev., 184 (1969), 1231–1260.CrossRefMathSciNetGoogle Scholar
  10. [BNR]
    H. L. Berk, W. M. Nevins and K. V. Roberts: New Stokes’ line in WKB theory, J. Math. Phys., 23 (1982), 988–1002.MATHCrossRefMathSciNetGoogle Scholar
  11. [CH]
    R. Courant and D. Hilbert: Methods of Mathematical Physics, II, Inter-science, 1962.Google Scholar
  12. [DDP]
    E. Delabaere, H. Dillinger and F. Pham: Exact semi-classical expansions for one dimensional quantum oscillators, J. Math. Phys., 38 (1997), 6126–6184.MATHCrossRefMathSciNetGoogle Scholar
  13. [H]
    L. Hörmander: Fourier integral operators I, Acta Math., 127 (1971), 79–183.MATHCrossRefMathSciNetGoogle Scholar
  14. [Ho]
    N. Honda: Toward the complete description of the Stokes geometry, in prep.Google Scholar
  15. [HLO]
    C. J. Howls, P. J. Langman and A. B. Olde Daalhuis: On the higher-order Stokes phenomenon, Proc. R. Soc. Lond. A, 460 (2004), 2285–2303.MATHCrossRefMathSciNetGoogle Scholar
  16. [KKNT]
    T. Kawai, T. Koike, Y. Nishikawa and Y. Takei: On the Stokes geometry of higher order Painlevé equations, Astérisque, No. 297, 2004, pp. 117–166.MathSciNetGoogle Scholar
  17. [KT]
    T. Kawai and Y. Takei: Algebraic Analysis of Singular Perturbation Theory, Iwanami, Tokyo, 1998. (In Japanese; English translation, AMS, 2005)Google Scholar
  18. [NY]
    M. Noumi and Y. Yamada: Higher order Painlevé Equations of type A l(1), Funkcial. Ekvac., 48 (1998), 483–503.MathSciNetGoogle Scholar
  19. [P]
    F. Pham: Resurgence, quantized canonical transformations, and multi-instanton expansions, Algebraic Analysis, vol. II, Acad. Press, 1988, pp. 699–726.MathSciNetGoogle Scholar
  20. [Sa1]
    S. Sasaki: On the role of virtual turning points in the deformation of higher order linear ordinary differential equations, RIMS Koukyuuroku, (ISSN 1880-2818), No. 1433, 2005, pp. 27–64. (In Japanese.)Google Scholar
  21. [Sa2]
    S. Sasaki: — II — On a new Stokes curve in Noumi-Yamada system, ibid., pp. 65–109. (In Japanese.)Google Scholar
  22. [SKK]
    M. Sato, T. Kawai and M. Kashiwara: Microfunctions and pseudo-differential equations, Lect. Notes in Math., No. 287, Springer, 1973, pp. 265–529.MathSciNetGoogle Scholar
  23. [Sh]
    A. Shudo: A recipe for finding Stokes geometry in quantized Hénon map, RIMS Koukyuuroku, (ISSN 1880-2818), No. 1433, 2005, pp. 110–118.Google Scholar
  24. [S]
    H. J. Silverstone: JWKB connection-formula problem revisited via Borel summation, Phys. Rev. Lett., 55 (1985), 2523–2526.CrossRefMathSciNetGoogle Scholar
  25. [T]
    Y. Takei: Exact WKB analysis, and exact steepest descent method, — A sequel to “Algebraic analysis of singular perturbations”, Sûgaku, 55 (2003), 350–367. (In Japanese. Its English translation will appear in Sûgaku Expositions.)MathSciNetGoogle Scholar
  26. [V]
    A. Voros: The return of the quartic oscillator. The complex WKB method, Ann. Inst. Henri Poincaré, 39 (1983), 211–338.MATHMathSciNetGoogle Scholar
  27. [Z]
    J. Zinn-Justin: Instantons in quantum mechanics: Numerical evidence for a conjecture, J. Math. Phys., 25 (1984), 549–555.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Takashi Aoki
    • 1
  • Naofumi Honda
    • 2
  • Takahiro Kawai
    • 3
  • Tatsuya Koike
    • 4
  • Yukihiro Nishikawa
    • 5
  • Shunsuke Sasaki
    • 6
  • Akira Shudo
    • 7
  • Yoshitsugu Takei
    • 3
  1. 1.School of Science and EngineeringKinki UniversityHigashi-OsakaJapan
  2. 2.Department of Mathematics, Graduate School of ScienceHokkaido UniversitySapporoJapan
  3. 3.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  4. 4.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  5. 5.Hitachi Ltd.TokyoJapan
  6. 6.Mitsubishi UFJ Securities Co. Ltd.TokyoJapan
  7. 7.Department of Physics, Graduate School of ScienceTokyo Metropolitan UniversityHachiojiJapan

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