Resurgent deformations for an ordinary differential equation of order 2
Résumé
We consider the differential equation (d2∕dx2)Φ(x) = (Pm(x)∕x2)Φ(x) in the complex field, where Pm is a monic polynomial function of order m. We investigate the asymptotic and resurgent properties of the solutions at infinity, focusing in particular on the analytic dependence of the Stokes–Sibuya multipliers on the coefficients of Pm. Taking into account the nontrivial monodromy at the origin, we derive a set of functional equations for the Stokes–Sibuya multipliers, and show how these relations can be used to compute the Stokes multipliers for a class of polynomials Pm. In particular, we obtain conditions for isomonodromic deformations when m = 3