General comparison theorems for the Klein-Gordon equation in d dimensions

We study bound-state solutions of the Klein-Gordon equation $ \varphi^{\prime\prime}(x) =[m^{2}-(E-v f(x))^{2}] \varphi(x)$, for bounded vector potentials which in one spatial dimension have the form $ V(x)=v f(x)$, where $ f(x)\le 0$ is the shape of a finite symmetric central potential that is monotone non-decreasing on $ [0, \infty)$ and vanishes as $ x\rightarrow\infty$. Two principal results are reported. First, it is shown that the eigenvalue problem in the coupling parameter v leads to spectral functions of the form $ v= G(E)$ which are concave, and at most uni-modal with a maximum near the lower limit $ E=-m$ of the eigenenergy $ E\in (-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\ge 2$, a comparison theorem is proven, to the effect that if two potential shapes are ordered $ f_{1}(r) \leq f_{2}(r)$, then so are the corresponding pairs of spectral functions $ G_{1}(E)\leq G_{2}(E)$ for each of the existing eigenvalues. These results remove the restriction to positive eigenvalues necessitated by earlier comparison theorems for the Klein-Gordon equation.