Another approach to WKB analysis

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.