  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.
