On the WKB-theoretic structure of a Schrödinger operator with a merging pair of a simple pole and a simple turning point

Abstract

A Schrödinger equation with a merging pair of a simple pole and a simple turning point (called MPPT equation for short) is studied from the viewpoint of exact Wentzel-Kramers-Brillouin (WKB) analysis. In a way parallel to the case of merging-turning-points (MTP) equations, we construct a WKB-theoretic transformation that brings an MPPT equation to its canonical form (the $\infty$-Whittaker equation in this case). Combining this transformation with the explicit description of the Voros coefficient for the Whittaker equation in terms of the Bernoulli numbers found by Koike, we discuss analytic properties of Borel-transformed WKB solutions of an MPPT equation.