Exact Solutions for the Singularly Perturbed Riccati Equation and Exact WKB Analysis

The singularly perturbed Riccati equation is the first-order nonlinear ODE $hbar partial_x f = af^2 + bf + c$ 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 halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., 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 Schrodinger equation with a rational potential.