On the Role of Differentiation Parameter in a Bound State Solution of the Klein-Gordon Equation

Recently, the bound state solutions of a confined Klein-Gordon particle under the mixed scalar-vector generalized symmetric Woods-Saxon potential in one spatial dimension have been investigated. The obtained results reveal that in the spin symmetric limit discrete spectrum exists, while in the pseudo-spin symmetric limit it does not. In this manuscript, new insights and information are given by employing an analogy of the variational principle. The role of the difference of the magnitudes of the vector and scalar potential energies, namely the differentiation parameter, on the energy spectrum is examined. It is observed that the differentiation parameter determines the measure of the energy spectrum density by modifying the confined particle's mass-energy in addition to narrowing the spectrum interval length.