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. {\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.