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

TL;DR

This paper applies a modified Newton's method to transform a complex Schrödinger equation with Gevrey series coefficients into a canonical form, advancing the analysis of factorially divergent series in quantum mechanics.

Contribution

It introduces a novel application of Newton's method to handle factorially divergent series in the context of complex WKB analysis, generalizing previous results.

Findings

Successfully transforms Schrödinger equation to canonical form

Handles factorially divergent power series in $h$

Extends prior work by Aoki, Kawai, and Takei

Abstract

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.