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.

arXiv comment11 pages, 10 figures and 1 table