An example of Newton's method for an equation in Gevrey series

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Getmanenko, Alexander^1,2 

[1] Univ Tokyo, Kavli IPMU, Tokyo 1138654, Japan

[2] UPMC, Math Inst Jussieu, Paris, France

关键词: Anharmonic oscillator;    Complex WKB;   

DOI  :  10.1016/j.jde.2014.09.002

来源: Elsevier

【 摘 要 】

In the context of complex WKB analysis, we discuss a one-dimensional Schrodinger equation -h(2)partial derivative(2)(x) f(x, h) +[Q(x) + hQ(1)(x, h)] f(x, h)=0, h -> 0, where Q(x), Q(1)(x, h) are analytic near the origin x = 0, Q(0) = 0, and Q(1)(x, h) is a factorially divergent power series in h. We show that there is a change of independent variable y = y(x, h), analytic near x = 0 and factorially divergent with respect to h, that transforms the above Schrodinger equation to a canonical form. The proof goes by reduction to a mildly nonlinear equation on y(x, h) and by solving it using an appropriately modified Newton's method of tangents. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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