Highly-damped quasi-normal frequencies for piecewise Eckart potentials

TL;DR

This paper investigates the conditions under which highly-damped quasi-normal frequencies follow a linear pattern, revealing that the rationality of exponential falloff ratios determines this behavior, with implications for black hole physics.

Contribution

It develops an analytic quantization condition for highly-damped quasi-normal frequencies in a piecewise Eckart potential, highlighting the role of exponential falloff ratios.

Findings

The linear pattern of frequencies is generic but not universal.

Rational ratios of exponential falloff rates lead to the linear frequency pattern.

Implications for black hole physics and damped modes are discussed.

Abstract

Highly-damped quasi-normal frequencies are very often of the form omega_n = (offset) + i n (gap). We investigate the genericity of this phenomenon by considering a model potential that is piecewise Eckart (piecewise Poeschl-Teller), and developing an analytic "quantization condition" for the highly-damped quasi-normal frequencies. We find that this omega_n = (offset) + i n (gap) behaviour is generic but not universal, with the controlling feature being whether or not the ratio of the rates of exponential falloff in the two asymptotic directions is a rational number. These observations are of direct relevance to any physical situation where highly-damped quasi-normal modes (damped modes) are important --- in particular (but not limited to) to black hole physics, both theoretical and observational.