Comparison theorems for the Klein-Gordon equation in d dimensions

Two comparison theorems are established for discrete eigenvalues of the Klein-Gordon equation with an attractive central vector potential in d >= 1 dimensions. (I) If \psi_1 and \psi_2 are node-free ground states corresponding to positive energies E_1 >= 0 and E_2 >= 0, and V_1® <= V_2® <= 0, then it follows that E_1 <= E_2. (II) If V(r,a) depends on a parameter a \in(a_1,a_2), V(r,a) <= 0, and E(a) is any positive eigenvalue, then \partial V/\partial a >= 0 ==> E’(a) >= 0 and \partial V/\partial a <= 0 ==> E’(a) <= 0.