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

In the context of complex WKB analysis, we discuss a one-dimensional Schr\"odinger equation $ -h^2\partial_x^2 f(x,h) + [Q(x)+hQ_1(x,h)]f(x,h) =0, \ \ \ h\to 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 Schr\"odinger 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.