Uniform Asymptotic Approximation Method with P\"oschl-Teller Potential

TL;DR

This paper develops a uniform asymptotic approximation method for second-order differential equations with two turning points, providing accurate analytical solutions for the P"oschl-Teller potential relevant in cosmology and black hole physics.

Contribution

It introduces a uniform asymptotic approximation approach for differential equations with two turning points, specifically applied to the P"oschl-Teller potential, with quantified error bounds.

Findings

Approximate solutions have errors generally less than 10%.

The method can be extended to higher-order approximations for increased accuracy.

Analytical solutions are applicable to cosmological perturbations and black hole quasi-normal modes.

Abstract

In this paper, we study analytical approximate solutions of the second-order homogeneous differential equations with the existence of only two turning points (but without poles), by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the P\"oschl-Teller (PT) potential, for which analytical solutions are known. Depending on the values of the parameters involved in the PT potential, we find that the upper bounds of the errors of the approximate solutions in general are $\lesssim 0.15\% \sim 10\% $, to the first-order approximation of the UAA method. The approximations can be easily extended to high-order, with which the errors are expected to be much smaller. Such obtained analytical solutions can be used to study cosmological perturbations in the framework of quantum cosmology, as well as quasi-normal modes of black holes.