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

TL;DR

This paper develops a method to find exact solutions for the singularly perturbed Riccati equation using Borel-Laplace summation, and applies it to establish the existence of exact WKB solutions for complex Schrödinger equations.

Contribution

It introduces a rigorous approach to construct exact solutions of the Riccati equation via Borel summation and applies this to WKB analysis of Schrödinger equations with rational potentials.

Findings

Existence and uniqueness of exact solutions with prescribed asymptotics.

Construction of solutions using Borel-Laplace method.

Application to WKB solutions for complex Schrödinger equations.

Abstract

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 Schr\"odinger equation with a rational potential.