On the Structure of Higher Order Simple-Pole Type Operators in Exact WKB Analysis

Abstract

We introduce an appropriate class of higher order linear ordinary differential operators with a large parameter and with simple poles in their coefficients, and study their structure from the viewpoint of exact WKB analysis, i.e., the WKB analysis based on the Borel resummation. Our main result is that an operator in the class is expressed as a product of two operators Q and R, with Q being irrelevant to the Stokes geometry and with R being a second order simple-pole type operator studied by Koike ([Ko1], [Ko2], [Ko3]). This decomposition theorem gives us a connection formula for WKB solutions near the simple pole in question, and it is used to explain the background mechanism of some intriguing phenomenon we encounter in the study of a particular third order operator. Some discussions on the scope of the future development of the theory are included.