On the mapping between bound states and black hole quasinormal modes via analytic continuation: a spectral instability perspective

TL;DR

This paper examines the connection between bound states and quasinormal modes in black hole perturbation theory, revealing the limits of spectral mapping and the influence of perturbation placement on spectral stability.

Contribution

It provides a detailed analysis of spectral instability, demonstrating how analytic continuation affects the mapping between bound states and quasinormal modes in specific potential models.

Findings

Spectral mapping reliability extends beyond the analytic continuation validity domain.

Perturbations near the potential extremum allow for accurate continuation to quasinormal frequencies.

Asymptotic perturbations lead to deformed spectra with no clear quasinormal mode correspondence.

Abstract

In this work, we investigate the relation between bound states and quasinormal modes within black hole perturbation theory in the context of spectral instability. Our analysis indicates that the reliability of such spectral mapping stretches beyond the domain of validity of the analytic continuation employed to connect the perturbative bound-state problem to the corresponding open-system dynamics. However, for the numerical scheme proposed by V\"olkel to work, the transformations of the metric parameters must be carried out in a region where the underlying Taylor expansion is convergent. As analytically accessible explicit examples, we explore the perturbed delta-function and P\"oschl-Teller potential barriers. For the latter, we construct two distinct perturbative setups for which the convergence of the series expansion involved in the perturbation theory can be rigorously controlled. When the deformation is placed near the potential's extremum, the resulting corrections to the bound-state energies can be analytically continued to yield perturbed quasinormal frequencies, in agreement with known semi-analytic results. In contrast, when the perturbation is localized asymptotically far from the compact object, the bound states are only mildly modified and are accurately described by a perturbative expansion to the first order. However, the associated analytic continuation yields a strongly deformed spectrum that shows no clear connection to the quasinormal modes. These findings contribute to the effort to scrutinize the conditions under which bound states faithfully encode quasinormal spectra and to shed light on the underlying physics of black hole spectral instability.