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Recently, scattering of a Klein-Gordon particle in the presence of mixedscalar-vector generalized symmetric Woods-Saxon potential was investigated forthe spin symmetric and the pseudo-spin symmetric limits in one spatialdimension. In this manuscript, the bound state solutions of the Klein-Gordonequation with mixed scalar-vector generalized symmetric Woods-Saxon potentialare examined analytically within the framework of spin and pseudo-spin symmetrylimits. We prove that the occurrence of bound state energy spectrum exists onlyin the spin symmetric limit, while in the pseudo-spin symmetric limit, thebound state spectrum does not exist. Besides the theoretical proof, theNewton-Raphson numerical methods are used to calculate the bound state energyspectra of a neutral Kaon particle, confined in a generalized symmetricWoods-Saxon potential, energy well constituted with repulsive or attractivesurface interactions, for the spin and pseudo-spin symmetric limits,respectively. Numerical results are consistent with the non-existence of thebound state energy spectrum in the pseudo-spin symmetric limit.

the european physical journal plus

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