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
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.
- W. Balser, From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics, vol. 1582, Springer-Verlag, 1994.
- S. Bodine, R. Schäfke, On the summability of formal solutions in Liouville-Green theory, J. Dynam. Control Systems 8 (2002), 371-398.
- E. Delabaere, F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré 71 (1999), 1-94.
- 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.
- T. Kawai, Y. Takei, Algebraic Analysis of Singular Perturbation Theory, Translations of Mathematical Monographs,Vol. 227, Amer. Math. Soc., 2005.
- T. Koike, R. Schäfke, On the Borel summability of WKB solutions of Schrödinger equations with polynomial potentials and its applications, in preparation.
- R. Schäfke, private communication.
- K. Suzuki, On multisummable WKB solutions of a certain ordinary differential equation of singular perturbation type, Master-thesis, Kyoto University, 2012.
- K. Suzuki, Y. Takei, Exact WKB analysis and multisummability - A case study , RIMS Kôkyûroku 1861 (2013), 146-155.
- 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.
