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

In the context of complex WKB analysis, we discuss a one-dimensional Schrödinger equation−h2∂x2f(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.