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

Abstract

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

Paper has been reorganised; notation, terminology, and theorem statements made clearer. Essential content unchanged

References in corpus (1)

• On the Borel summability of WKB solutions of certain Schrödinger-type differential equations

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