Exact WKB Analysis of Schrödinger Equations with a Stokes Curve of Loop Type

Abstract

Stokes phenomena with respect to a large parameter are investigated for Shrödinger-type ordinary di¤erential equations having a Stokes curve of loop-type. For this purpose, we employ a Bessel-type equation as a canonical form and compute the Voros coe‰cient of the equation. Combining the formula describing the Stokes automorphism for the Voros coe‰cient and the formal coordinate transformation connecting the Shrödinger-type equation and the Bessel-type equation, we have some formulas describing the action of alien derivatives and Stokes automorphism for WKB solutions of the Shrödinger-type equation.