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

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.