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

Abstract

AbstractThe singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation $\hbar \partial _x f = af^2 + bf + c$ in the complex domain where $\hbar $ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent $\hbar $ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.