A Note on Zero Free Regions of the Stokes Multipliers for Second Order Ordinary Differential Equations with Cubic Polynomial Coefficients

MR2918148 Tatsuo NISHITANI Tatsuo NISHITANI A Note on Zero Free Regions of the Stokes Multipliers for Second Order Ordinary Differential Equations with Cubic Polynomial Coefficients Funkcialaj Ekvacioj. Serio Internacia 54 2011 473--483 http://fe.math.kobe-u.ac.jp/FE/FullPapers/54-3/54_473.pdf http://www.ams.org/mathscinet-getitem?mr=MR2918148 In this note, by a completely elementary argument, we give a rough estimate about zero free regions of the Stokes multipliers for second order differential equations with cubic polynomial coefficients. We also prove that the Stokes multipliers indeed has zeros outside this zero free regions by resorting to some classical results about spectral problem for anharmonic oscillators. Zero free region, Stokes multiplier, Anharmonic oscillator. Primary: 34M; Secondary: 34M40. 54-473 2011 A Note on Zero Free Regions of the Stokes Multipliers for Second Order Ordinary Differential Equations with Cubic Polynomial Coefficients Tatsuo NISHITANI Tatsuo NISHITANI

1 Bernardi, E.; Nishitani, T. On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness J. Anal. Math. 105 2008 197--240 MR2438425

2 Bernardi, E.; Nishitani, T., On the Cauchy problem for noneffectively hyperbolic operators, the Gevrey 3 well-posedness, to appear in J. Hyperbolic Differ. Equ.

3 Caliceti, E.; Graffi, S.; Maioli, M. Perturbation theory of odd anharmonic oscillators Comm. Math. Phys. 75 1980 51--66 MR0581569

4 Caliceti, E. Distributional Borel summability of odd anharmonic oscillators J. Phys. A 33 2000 3753--3770 MR1767037

5 Greene, R. E.; Krantz, S. G. Function Theory of One Complex Variable Graduate Studies in Mathematics, 40, American Mathematical Society, Providence, RI 2006 MR2215872 6 Sibuya, Y. Global Theory of a Second Order Linear Ordinary Differential Equations with a Polynomial Coefficients North-Holland Mathematics Studies, Vol. 18, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York 1975 MR0486867

7 Trinh, D.Tai On the simpleness of zeros of Stokes multipliers J. Differential Equations 223 2006 351--366 MR2214939