2013/04/24 by Alexander Getmanenko, Getmanenko, Alexander
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA
paper · pdf · doi:10.48550/arxiv.1304.6621
arxiv created 2013/04/24 · arxiv updated 2013/04/25
In the context of complex WKB analysis, we discuss a one-dimensional Schrödinger equation -h2∂x2 f(x,h) + [Q(x)+hQ1(x,h)]f(x,h) =0, h→ 0, where Q(x), Q1(x,h) are analytic near the origin x=0, Q(0)=0, and Q1(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 Schrödinger 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. Our result generalizes that of Aoki, Kawai, and Takei.