Another approach to WKB analysis

TL;DR

This paper introduces an elementary approach to analyzing the WKB equation, a singular perturbation problem, using advanced calculus and differential equations without relying on assumptions about turning points or Stokes curves.

Contribution

It presents a new, straightforward method for studying the WKB equation, expanding analysis beyond traditional assumptions using elementary mathematical tools.

Findings

Developed an elementary method for WKB analysis.

Applicable without assuming simple turning points or Stokes curve connections.

Provides new insights into the structure of solutions to the WKB equation.

Abstract

A singular perturbation problem called WKB equation (Eq) $h^2u(x,h)-Q(x)u(x,h)=0$ is studied. $h>0$ is a small parameter. Investigation of (Eq) has long history. Recently it has developed by a new method named "Exact WKB Analysis" based on Borel resummation method and new analytic results. Here we study (Eq) by another elementary method. We only apply advanced calculus and the theory of differential equations to (Eq). We neither assume turning points are simple nor there is no Stokes curve that connects two turning points.