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We first study bound-state solutions of the Klein--Gordon equation φ"(x)=[m2 - (E-V(x))2] φ(x), for vector potentials which in one spatial dimension have the form V(x) = vf(x), where f(x)≤0 is the shape of a finite symmetric central potential that is monotone non-decreasing on [0, ∞) and vanishes as x⟶∞, and v>0 is the coupling parameter.
We characterize the graph of spectral functions of the form v= G(E) which represent solutions of the eigen-problem in the coupling parameter v for a given E: they are concave, and at most uni-modal with a maximum near the lower limit E = -m of the energy E ∈ (-m, m). This formulation of the spectral problem immediately extends to central potentials in d > 1 spatial dimensions. Secondly, for each of the dimension cases, d=1 and d≥ 2, a comparison theorem is proven, to the effect that if two potential shapes are ordered f1(r) ≤ f2(r), then so are the corresponding pairs of spectral functions G1(E) ≤ G2(E) for each of the existing eigenvalues. These results remove the restriction to positive energies necessitated by earlier comparison theorems for the Klein--Gordon equation by Hall and Aliyu. Corresponding results are obtained when scalar potentials S(x) are also included.
We then weaken the condition for the ground states by proving that if ∫x0 [f2(t) - f1(t)]φ(t) dt≥0, the corresponding coupling parameters remain ordered, where φi= 1, or φi is the bound state solution of the Klein--Gordon equation with potential Vi, i = 1, 2. These results are valid for any energy E ∈ (-m, m)., but they are restricted to the ground states.
We finally present a complete recipe for finding upper and lower spectral bounds for both bounded and unbounded potentials, and we exhibit specific result for the applications for the Woods-Saxon, Gaussian, sech-squared, and Yukawa potentials in dimensions d = 1 and d = 3.