Microlocal Analysis of Fixed Singularities of WKB Solutions of a Schrödinger Equation with a Merging Triplet of Two Simple Poles and a Simple Turning Point

Abstract

We first show that the WKB-theoretic canonical form of an M2P1T (merging two poles and one turning point) Schrödinger equation is given by the algebraic Mathieu equation. We further show that, in analyzing the structure of WKB solutions of a Mathieu equation near fixed singular points relevant to simple poles of the equation, we can focus our attention on the pole part of the equation so that we may reduce it to the Legendre equation. The Borel transformation of WKB-theoretic transformations thus obtained gives rise to microdifferential relations, which lead to the microlocal analysis of the Borel transformed WKB solutions of an M2P1T equation near their fixed singular points. The fully detailed account of the results will be given in Kamimoto et al. (Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point—its relevance to the Mathieu equation and the Legendre equation, 2011).