Bound-State Resonances of Schwarzschild-de Sitter Black Holes: Analytic Treatment
Inspired by Mashhoon's framework connecting black hole quasi-normal modes (QNMs) to bound-state resonances in inverted potentials, V$\ddot{\text{o}}$lkel'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 wavefunctions 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 $Λ\rightarrow 0$ 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 resonance eigenfunctions 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 $Λ$ 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 half-bound states and the number of bound-state resonance energy levels.