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

Abstract

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 Schrödinger equations. Our method is motivated by and has applications to the analysis of linear wave equations in the geometry of a rotating black hole.

Absence of zeros and asymptotic error estimates for airy and parabolic cylinder functions

“…Our method for proving such a spectral representation requires detailed information on the eigensolutions. This information can be obtained by "glueing together" approximate solutions and controlling the error using the methods in [3]. One method for obtaining such approximate solutions is to solve the Sturm-Liouville equation…”

“…with quantitative error bounds. In order to apply the methods in [3], we also need to control the function y := φ ′ /φ, which solves the associated Riccati equation…”