Quasi-normal frequencies: Key analytic results

TL;DR

This paper surveys and develops new analytic results on quasi-normal frequencies, providing key estimates and bounds that are relevant across black hole physics, quantum field theory, and condensed matter physics.

Contribution

It offers new analytic formulas and estimates for QNFs, enhancing understanding and comparison with existing literature, and explores applications of the Lambert W function in physics.

Findings

New QNF results and estimates are derived.

Provides analytic formulas for comparison with literature.

Highlights relevance of Lambert W function in physics applications.

Abstract

The study of exact quasi-normal modes [QNMs], and their associated quasi-normal frequencies [QNFs], has had a long and convoluted history - replete with many rediscoveries of previously known results. In this article we shall collect and survey a number of known analytic results, and develop several new analytic results - specifically we shall provide several new QNF results and estimates, in a form amenable for comparison with the extant literature. Apart from their intrinsic interest, these exact and approximate results serve as a backdrop and a consistency check on ongoing efforts to find general model-independent estimates for QNFs, and general model-independent bounds on transmission probabilities. Our calculations also provide yet another physics application of the Lambert W function. These ideas have relevance to fields as diverse as black hole physics, (where they are related to the damped oscillations of astrophysical black holes, to greybody factors for the Hawking radiation, and to more speculative state-counting models for the Bekenstein entropy), to quantum field theory (where they are related to Casimir energies in unbounded systems), through to condensed matter physics, (where one may literally be interested in an electron tunelling through a physical barrier).