Quasinormal modes and the analytical continuation of non-self-adjoint operators

TL;DR

This paper reviews the analytical continuation method for quasinormal modes, demonstrating its limitations in fully determining QNMs, especially for non-self-adjoint operators, with applications to black hole physics.

Contribution

It reveals fundamental limitations of the analytical continuation method in computing complete QNM spectra for non-self-adjoint operators.

Findings

Analytical continuation cannot determine all QNMs from bound states.

Many QNMs are analytically continued solutions outside the self-adjoint domain.

Application to BTZ black holes illustrates these issues.

Abstract

We briefly review the analytical continuation method for determining quasinormal modes (QNMs) and the associated frequencies in open systems. We explore two exactly solvable cases based on the P\"oschl-Teller potential to show that the analytical continuation method cannot determine the full set of QNMs and frequencies of a given problem starting from the associated bound state problem in Quantum Mechanics. The root of the problem is that many QNMs are the analytically continued counterparts of solutions that do not belong to the domain where the associated Schr\"odinger operator is self-adjoint, challenging the application of the method for determining full sets of QNMs. We illustrate these problems through the physically relevant case of BTZ black holes, where the natural domain of the problem is the negative real line.