Relativistic bound-state solutions for a non-central Schi{\"o}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.