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

Abstract

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 on each hemisphere with principal boundary conditions and treat the radial equation with the Greene--Aldrich approximation, obtaining Jacobi-polynomial wavefunctions and an implicit relativistic quantization condition. With this boundary realization, the equatorial inverse-square singularity splits configuration space into reflection-related sectors with identical angular spectra. Reducing the barrier to zero while retaining the equatorial wall selects the odd-reflection spectral branch of the regular full-domain problem; a single hemisphere has no intrinsic parity. The nonrelativistic limit recovers the corresponding Schr{ö}dinger formulation. State-resolved nonrelativistic and relativistic benchmarks quantify the centrifugal error. In a dimensionless relativistic example, the relative error in the remaining binding exceeds unity near threshold although the absolute energy error decreases. For NaH and Na${}_2$, low-lying central-radial vibrational energies agree reasonably with spectroscopic reference values, whereas the global radial shape limits near-dissociation accuracy and gives an incorrect NaH bound-state count.

12 pages, 4 figures