Comment on "On the bound states of the Schwarzschild black hole" by S. H. V\"olkel: A Reassessment of the Bound-State Analogy

TL;DR

This paper critically refutes a recent claim that black hole quasinormal modes can be derived from bound states, highlighting mathematical errors and inconsistencies in the proposed spectral mapping and boundary conditions.

Contribution

It provides a detailed reassessment demonstrating that the bound-state analogy for black hole QNMs is invalid due to mathematical and physical inaccuracies.

Findings

The spectral mapping used is mathematically invalid.

The boundary conditions are misapplied, leading to incorrect results.

The inversion method fails to capture the complex nature of black hole resonances.

Abstract

This comment critically examines the recent proposal by S.~H.~V\"olkel [Phys. Rev. Lett., arXiv:2505.17186], which asserts that the quasinormal mode (QNM) spectrum of Schwarzschild black holes can be reconstructed from bound states of an inverted Regge--Wheeler potential. We demonstrate that this claim rests on a mathematically invalid spectral mapping and a misapplication of boundary conditions that define QNMs. Through a detailed figure-by-figure analysis, we expose deep inconsistencies in the numerical results and their physical interpretations. The inversion method, inspired by Mashhoon, is shown to fail in capturing the non-Hermitian, complex nature of black hole resonances. We contrast this with the rigorously established asymptotic structure of QNMs derived by Hod and others, concluding that the bound-state framework offers no reliable insight into black hole spectroscopy.