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

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. 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 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 ${{E}_{n}{}}_{n=0}^{n=\ensuremath{\infty}}$ 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.