On the Complex WKB Analysis for a Second Order Linear O.D.E. with a Many-Segment Characteristic Polygon

Abstract

Asymptotics of ODE appearing in the turning point problems can be characterized literally by its characteristic polygon. The Airy equation has a one-segment characteristic polygon. Fedoryuk ([4]), Nakano ([8], [12]), Nakano et al. ([13]), and Roos ([17], [18]) studied ODE's with a several-segement one. The more segments, the more complicated asymptotics. Here, we study an ODE with a many-segment one. Firstly, the ODE is reduced to the simpler ODE's in some subdomains, and then the reduced ODE's have the WKB solutions as their asymptotic solutions. Secondly, two sets of the WKB solutions in the neighboring subdomains are related by a matching matrix. In our analysis the stretching-matching method is applied and the Stokes curves play an important role. How to get the Stokes curve configuration for the reduced ODE's is analyzed precisely.