The Treatment of the (3+1)-Dimensional Klein-Gordon Equation under Coulomb Interaction within the Framework of the Yang Model

We investigate the three-dimensional Klein–Gordon equation for a spinless particle in a Coulomb potential within the framework of a first-order phenomenological application of the Yang deformation, simultaneously incorporates a minimal measurable length and a minimal measurable momentum. Using the Hermitian realization of the deformed operators to first order in both deformation parameters, we derive the full radial equation and obtain the energy spectrum via first-order perturbation theory. In the free-particle limit, we recover the exact discrete spectrum of the Yangconfined free particle. The [Formula: see text]-correction is negative and grows quadratically with the principal quantum number, reflecting confinement to a sphere of radius [Formula: see text], while the [Formula: see text]-correction scales as [Formula: see text] and reflects smearing of the Coulomb singularity by the minimal length. The [Formula: see text]-correction is valid only for [Formula: see text], since the relevant expectation values diverge for s-states. The full spectrum is therefore perturbative and does not include the [Formula: see text] sector for the [Formula: see text]-correction. For the s-states [Formula: see text], we have shown that this divergence can be regularized by introducing a physical short-distance cutoff [Formula: see text]. Both corrections partially break the hydrogen degeneracy through their dependence on the angular-momentum quantum number. We analyze all relevant limiting cases and compare with existing results. In the non-relativistic limit, the binding energy reproduces the known three-dimensional hydrogen spectrum with Yang corrections up to a state-independent constant shift.