On the wall-crossing formula for quadratic differentials

Understand

We prove an analytic version of the Kontsevich-Soibelman wall-crossing formula describing how the number of finite-length trajectories of a quadratic differential jumps as the differential is varied.

• We characterize certain birational automorphisms of an algebraic torus appearing in this wall-crossing formula using Fock-Goncharov coordinates.
• As an application, we compute the Stokes automorphisms for the Voros symbols appearing in the exact WKB analysis of Schr\"odinger's equation.