Bound-state resonances of the Schwarzschild black hole: Analytic treatment

TL;DR

This paper analytically derives the infinite spectrum of bound-state resonances for the inverted Schwarzschild black-hole potential, confirming recent numerical results and deepening understanding of black hole quasinormal modes.

Contribution

The paper provides the first closed-form analytical formulas for the bound-state energy spectrum of the inverted Schwarzschild potential, aligning with recent numerical findings.

Findings

Analytical formulas for the bound-state energy spectrum are derived.

The analytical spectrum agrees well with recent numerical data.

The work enhances understanding of black hole quasinormal modes.

Abstract

Inspired by an earlier idea of Mashhoon, who suggested to relate the discrete quasinormal resonant modes of a black hole to the bound-state resonances of the corresponding inverted black-hole potential, V\"olkel [Phys. Rev. Lett. {\bf 134}, 241401 (2025)] has recently computed numerically, for the first time, the bound-state energy spectrum of the inverted Schwarzschild potential. Motivated by this intriguing work, in the present work we use {\it analytical} techniques in order to explore the physical and mathematical properties of the Schwarzschild bound-state resonances. In particular, we derive closed-form compact analytical formulas for the infinite spectrum $\{E_n\}_{n=0}^{n=\infty}$ of energy eigenvalues that characterize the inverted (binding) black-hole potential. Interestingly, it is explicitly shown that our analytically derived energy spectrum of the black-hole inverted potential agrees remarkably well with the corresponding numerical data that recently appeared in the physics literature.