Broken isospectrality in inverted black hole potentials and the emergence of Efimov scaling
Abstract
Are the inverted Regge-Wheeler and Zerilli potentials arising in Mashhoon's bound-state approach to black-hole quasinormal modes isospectral? Extending previous numerical and analytical studies of the inverted Regge-Wheeler potential, we present the first direct computation of the bound-state spectrum of the inverted Zerilli potential. Although Chandrasekhar proved that the original potentials have identical quasinormal-mode spectra, we show that their inverted counterparts have distinct bound-state spectra. We also demonstrate that potential inversion destroys their first-order Darboux relation, used by Glampedakis et al. to provide a simpler proof of isospectrality. A continuous interpolation between the potentials reveals substantially larger energy shifts in the interior than at the endpoints, indicating cancellations that suppress the endpoint spectral splitting. Explaining this behavior requires second-order perturbative contributions, even towards the eikonal limit. Despite these differences, both spectra exhibit asymptotic geometric scaling characteristic of Efimov physics, first discovered in three-boson systems. This scaling also applies to other inverted potentials with the same leading tail, including the scalar-field case studied here. Beyond their theoretical implications, our results raise questions for recently developed methods that compute quasinormal modes with high precision from numerical bound states.
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BibTeX
@misc{shahzad2026broken,
title = {{Broken isospectrality in inverted black hole potentials and the emergence of Efimov scaling}},
author = {Umer Shahzad and Sebastian H. Völkel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27102},
eprint = {2610.27102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27102},
}
RIS
TY - UNPB
TI - Broken isospectrality in inverted black hole potentials and the emergence of Efimov scaling
AU - Umer Shahzad
AU - Sebastian H. Völkel
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27102
UR - https://arxiv.org/abs/2610.27102
AB - Are the inverted Regge-Wheeler and Zerilli potentials arising in Mashhoon's bound-state approach to black-hole quasinormal modes isospectral? Extending previous numerical and analytical studies of the inverted Regge-Wheeler potential, we present the first direct computation of the bound-state spectrum of the inverted Zerilli potential. Although Chandrasekhar proved that the original potentials have identical quasinormal-mode spectra, we show that their inverted counterparts have distinct bound-state spectra. We also demonstrate that potential inversion destroys their first-order Darboux relation, used by Glampedakis et al. to provide a simpler proof of isospectrality. A continuous interpolation between the potentials reveals substantially larger energy shifts in the interior than at the endpoints, indicating cancellations that suppress the endpoint spectral splitting. Explaining this behavior requires second-order perturbative contributions, even towards the eikonal limit. Despite these differences, both spectra exhibit asymptotic geometric scaling characteristic of Efimov physics, first discovered in three-boson systems. This scaling also applies to other inverted potentials with the same leading tail, including the scalar-field case studied here. Beyond their theoretical implications, our results raise questions for recently developed methods that compute quasinormal modes with high precision from numerical bound states.
ER -