Opuscula Math. 35, no. 5 (2015), 775-802
http://dx.doi.org/10.7494/OpMath.2015.35.5.775

 
Opuscula Mathematica

On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations

Yoshitsugu Takei

Abstract. Using two concrete examples, we discuss the multisummability of WKB solutions of singularly perturbed linear ordinary differential equations. Integral representations of solutions and a criterion for the multisummability based on the Cauchy-Heine transform play an important role in the proof.

Keywords: exact WKB analysis, WKB solution, multisummability.

Mathematics Subject Classification: 34M60, 34E20, 34M30, 40G10.

Full text (pdf)

  1. W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics, vol. 1582, Springer-Verlag, 1994.
  2. S. Bodine, R. Schäfke, On the summability of formal solutions in Liouville-Green theory, J. Dynam. Control Systems 8 (2002), 371-398.
  3. E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré 71 (1999), 1-94.
  4. T.M. Dunster, D.A. Lutz, R. Schäfke, Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions, Proc. Roy. Soc. London, Ser. A 440 (1993), 37-54.
  5. T. Kawai, Y. Takei, Algebraic Analysis of Singular Perturbation Theory, Translations of Mathematical Monographs,Vol. 227, Amer. Math. Soc., 2005.
  6. T. Koike, R. Schäfke, On the Borel summability of WKB solutions of Schrödinger equations with polynomial potentials and its applications, in preparation.
  7. R. Schäfke, private communication.
  8. K. Suzuki, On multisummable WKB solutions of a certain ordinary differential equation of singular perturbation type, Master-thesis, Kyoto University, 2012.
  9. K. Suzuki, Y. Takei, Exact WKB analysis and multisummability - A case study , RIMS Kôkyûroku 1861 (2013), 146-155.
  10. A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. Inst. H. Poincaré 39 (1983), 211-338.
  • Yoshitsugu Takei
  • RIMS, Kyoto University, Kyoto 606-8502, Japan
  • Communicated by P.A. Cojuhari.
  • Received: 2014-03-31.
  • Revised: 2014-08-08.
  • Accepted: 2014-12-15.
  • Published online: 2015-04-27.
Opuscula Mathematica - cover

Cite this article as:
Yoshitsugu Takei, On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations, Opuscula Math. 35, no. 5 (2015), 775-802, http://dx.doi.org/10.7494/OpMath.2015.35.5.775

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.