The Bender - Wu analysis and the Voros theory

In our earlier paper ([AKT1]), by interpreting the formal transformation to the Airy equation near a simple turning point as the symbol of a microdifferential operator, we derived the Voros connection formula or, equivalently, the discontinuity function of a Borel transformed WKB solution at its movable singularities. In this paper we extend this approach to the two turning points problem; by constructing the formal transformation which brings a Schrödinger equation with two paired simple turning points that merge (i.e., a merging-turning-points equation or an MTP equation for short) to the Weber equation and by interpreting it as the symbol of a microdifferential operator, we reduce the analysis of an MTP equation to that of the Weber equation. Then, combining this transformation theory with the so-called “Sato’s conjecture” for the Weber equation, we obtain the discontinuity function of a Borel transformed WKB solution of an MTP equation at its fixed singularities.