Highly damped quasinormal frequencies for piecewise Eckart potentials

Abstract

Highly damped quasinormal frequencies are very often of the form ${\ensuremath{\omega}}_{n}=(\mathrm{\text{offset}})+in(\mathrm{\text{gap}})$. We investigate the genericity of this phenomenon by considering a model potential that is piecewise Eckart (piecewise P\"oschl-Teller), and developing an analytic ``quantization condition'' for the highly damped quasinormal frequencies. We find that this ${\ensuremath{\omega}}_{n}=(\mathrm{\text{offset}})+in(\mathrm{\text{gap}})$ behavior 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 quasinormal modes (damped modes) are important---in particular (but not limited to) to black hole physics, both theoretical and observational.