Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials
This work provides a rigorous error estimation framework for approximate solutions of the Riccati equation, which is relevant for analyzing wave equations in black hole spacetimes, but the contribution is incremental as it extends existing techniques.
The paper presents a method for obtaining rigorous error estimates for approximate solutions of the Riccati equation with real or complex potentials, using invariant region estimates. The method is illustrated on examples involving WKB and Airy solutions, with applications to linear wave equations in rotating black hole geometry.
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.