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

Nikita Nikolaev

Nikita Nikolaev

**14/8/20**Published in : arXiv:2008.06492

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 Schrödinger equation with a rational potential.