Geometry and Resurgence of WKB Solutions of Schr\"odinger Equations

Abstract

We prove that formal WKB solutions of Schr\"odinger equations on Riemann surfaces are resurgent. Specifically, they are Borel summable in almost all directions and their Borel transforms admit endless analytic continuation away from a discrete subset of singularities. Our approach is purely geometric, relying on understanding the global geometry of complex flows of meromorphic vector fields using techniques from holomorphic Lie groupoids and the geometry of spectral curves. This framework provides a fully geometric description of the Borel plane, Borel singularities, and the Stokes rays. In doing so, we introduce a geometric perspective on resurgence theory.