Inspired by Mashhoon’s framework connecting black-hole quasinormal modes to bound-state resonances in inverted potentials, Völkel’s recent numerical analysis of asymptotically flat Schwarzschild black holes revealed a counterintuitive phenomenon: highly excited bound states rapidly delocalize, become extremely weakly bound, and exhibit wave functions highly sensitive to far-field perturbations. To analytically explain this phenomenon and extend the investigation to Schwarzschild–de Sitter (SdS) black holes, we derive the characteristic equation for excited bound-state resonances in SdS spacetime and obtain compact closed-form analytical expressions for their resonance energies. In the $\mathrm{\Lambda }\to 0$ limit, our SdS-derived spectrum aligns perfectly with recent results for Schwarzschild black holes. We analytically demonstrate that the rapid and infinite delocalization of highly excited resonances is a universal feature of asymptotically flat Schwarzschild systems. More significantly, we prove that SdS black holes support only a finite number of bound-state resonance levels—in sharp contrast to the infinite spectrum of the asymptotically flat case. This finiteness implies an upper bound on the oscillatory domain of the bound-state resonances in SdS geometries, thereby preventing infinite delocalization and offering a fundamental distinction in the resonance structure of black holes in different asymptotic backgrounds. Surprisingly, we also find that delocalized half-bound states exist in SdS black holes when the $\mathrm{\Lambda }$ takes specific discrete values. This is a unique feature of SdS black holes and is absent in asymptotically flat Schwarzschild black holes. We also reveal the deep connection between the half-bound states and the number of bound-state resonance energy levels.

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• Classical black holes
• General relativity