On the Exact WKB Analysis of Second Order Linear Ordinary Differential Equations with Simple Poles

Abstract

on a neighborhood of a simple pole of the potential Q(x). Here and in what follows rj denotes a large parameter. Such a study of (1.1) has been completely done near a point where Q(x) is holomorphic and has a simple zero (i.e., near a simple turning point) or near a point where Q(x) has a double pole. (See [9], [1], [2], and references cited there.) However no exact WKB-theoretic study of (1.1) near a simple pole has ever been done. (See [4] and [8] for some nonexact WKB-theoretic study.) The difficulty arises from the fact that a simple pole plays also a role of turning points in the exact WKB analysis as it appears as a consequence of the confluence of a simple turning point and a double pole. (Cf. § 3 below.) In this paper we establish a connection formula for the Borel transform of WKB solutions of (1.1) near a simple pole of Q(x), using the canonical equation found in [5]. The plan of this paper is as follows: In §2, after introducing some basic notions and notations in the exact WKB analysis, we state our main theorem on the connection formula for the Borel transform of WKB solutions of (1.1) near a simple pole. In §3 we first recall the results in [5] which enable us to bring the equation (1.1) to the canonical form