An investigation of the bound-state solutions of the Klein-Gordon equation for the generalized Woods-Saxon potential in spin symmetry and pseudo-spin symmetry limits

Recently, scattering of a Klein-Gordon particle in the presence of mixed scalar-vector generalized symmetric Woods-Saxon potential was investigated for the spin symmetric and the pseudo-spin symmetric limits in one spatial dimension. In this paper, the bound-state solutions of the Klein-Gordon equation with mixed scalar-vector generalized symmetric Woods-Saxon potential are examined analytically within the framework of spin and pseudo-spin symmetry limits. We prove that the occurrence of the bound-state energy spectrum exists only in the spin symmetric limit, while in the pseudo-spin symmetric limit, the bound-state spectrum does not exist. Besides the theoretical proof, the Newton-Raphson numerical methods are used to calculate the bound-state energy spectra of a neutral kaon particle, confined in a generalized symmetric Woods-Saxon potential, energy well constituted with repulsive or attractive surface interactions, for the spin and pseudo-spin symmetric limits, respectively. Numerical results are consistent with the non-existence of the bound-state energy spectrum in the pseudo-spin symmetric limit.