Skip to main content
SpringerLink
Log in

The Bender-Wu Analysis and the Voros Theory

To the memory of the late Professor K. Yosida

  • Conference paper
ICM-90 Satellite Conference Proceedings

Abstract

In their pioneering work [BW], Bender and Wu presented the secular equation for anharmonic oscillators ([BW], (F.56)~(F.58) in p. 1257), which was supported by their ingenious WKB analysis. As we shall discuss in our forthcoming article, we can validate their conjecture by Voros’ epoch-making article [V]. At the same time several ideas contained in [BW] can be effectively employed to understand the Voros theory from the viewpoint of (micro)differential operators, and this is what we report here.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aoki, T., M. Kashiwara and T. Kawai: On a class of linear differential operators of infinite order with finite index. Adv. in Math., 62 (1986), 155–168.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bender, C.M. and T.T. Wu: Anharmonic oscillator. Phys. Rev., 184 (1969), 1231–1260.

    Article  MathSciNet  Google Scholar 

  3. Candelpergher, B., C. Nosmas and F. Pham: Résurgence et développements semiclassiques. To appear.

    Google Scholar 

  4. Ecalle, J.: Cinq applications des fonctions résurgentes. Prépublications d’Orsay, 84T62, Univ. Paris-Sud, 1984.

    Google Scholar 

  5. Gérard, C. and A. Grigis: Precise estimates of tunneling and eigenvalues near a potential barrier. J. Diff. Eq., 72 (1988), 149–177.

    Article  MATH  Google Scholar 

  6. Pham, F.: Resurgence, quantized canonical transformations, and multi-instant on expansions. Algebraic Analysis, vol. II, Academic Press, 1988, pp.699–726.

    Google Scholar 

  7. Pham, F.: A talk at Hayashibara Forum, 1990 (Okayama). In this volume.

    Google Scholar 

  8. Sato, M., T. Kawai and M. Kashiwara: Microfunctions and pseudo-differential equations. Lecture Notes in Math., 287, Springer, 1973, pp.265–529.

    MathSciNet  Google Scholar 

  9. Voros, A.: The return of the quartic oscillator. The complex WKB method. Ann. Inst. Henri Poincaré, 39 (1983), 211–338.

    MathSciNet  MATH  Google Scholar 

  10. Zinn-Justin, J.: Instantons in quantum mechanics: Numerical evidence for a conjecture. J. Math. Phys., 25 (1984), 549–555.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Faculty of Science and Technology, Kinki University, Higashi-Osaka 577, Japan

    Takashi Aoki

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan

    Takahiro Kawai

  3. Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan

    Yoshitsugu Takei

Authors
  1. Takashi Aoki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takahiro Kawai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoshitsugu Takei
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RIMS, Kyoto University, Sakyoku, Kyoto 606, Japan

    Masaki Kashiwara  & Tetsuji Miwa  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag Tokyo

About this paper

Cite this paper

Aoki, T., Kawai, T., Takei, Y. (1991). The Bender-Wu Analysis and the Voros Theory. In: Kashiwara, M., Miwa, T. (eds) ICM-90 Satellite Conference Proceedings. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68170-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-68170-0_1

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-70085-2

  • Online ISBN: 978-4-431-68170-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation