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

The singularly perturbed Riccati equation is the first-order nonlinear ODE $\mathrm{\hslash }{\mathrm{\partial }}_{x}f=a{f}^2+bf+c$ in the complex domain where $\mathrm{\hslash }$ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\mathrm{\hslash }\to 0$ in a halfplane. These exact solutions are constructed using the Borel-Laplace method; i.e., they are Borel summations of the formal divergent $\mathrm{\hslash }$-power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schr"odinger equation with a rational potential.