EXACT SOLUTIONS FOR THE SINGULARLY PERTURBED RICCATI EQUATION AND EXACT WKB ANALYSIS

The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation h partial derivative(x)f = af(2) + bf + c in the complex domain where h is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as h -> 0 in a half-plane. These exact solutions are constructed using the Borel-Laplace method; that is, they are Borel summations of the formal divergent h-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schro spexpressioncing diexpressioneresis dinger equation with a rational potential.