Relativistic bound-state solutions for a non-central Schi{ö}berg-type hyperbolic potential with double ring-shaped angular terms

Singular interactions can reshape quantum spectra by changing admissible wavefunction domains. In this study, we investigate this mechanism in the Klein–Gordon equation for a non-central Schiöberg-type hyperbolic potential with double ring-shaped angular barriers. Equal scalar and vector couplings separate the radial and angular dynamics. We solve the angular equation exactly and treat the radial equation with the Greene–Aldrich approximation, obtaining Jacobi-polynomial wavefunctions and an implicit relativistic quantization condition. The equatorial inverse-square singularity splits configuration space into reflection-related sectors with identical angular spectra. Reducing the barrier to zero within one sector retains only odd-parity states, unlike the regular full-domain problem, which restores both parities. The nonrelativistic limit recovers the Schr{ö}dinger spectrum. For NaH and Na${}_2$, low-lying vibrational energies agree reasonably with experiment, whereas near-dissociation deviations reveal asymptotic limitations.