Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials

TL;DR

This paper introduces a method for rigorously estimating errors in approximate solutions to the Riccati equation with real or complex potentials, using invariant region estimates and applications to wave equations in black hole geometries.

Contribution

It develops a novel approach for error estimation in Riccati equations, combining invariant region techniques with WKB and Airy solutions, applicable to complex potentials and black hole wave analysis.

Findings

Invariant region estimates effectively bound complex Riccati solutions.

The method applies to approximate solutions from WKB and Airy methods.

Potential applications include analyzing wave equations in rotating black hole spacetimes.

Abstract

A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by glueing together WKB and Airy solutions of corresponding one-dimensional Schrodinger equations. Our method is motivated by and has applications to the analysis of linear wave equations in the geometry of a rotating black hole.