Comparison theorems for the Klein-Gordon equation in d dimensions
Abstract
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 1 and 2 are node-free ground states corresponding to positive energies E1 >= 0 and E2 >= 0, and V1(r) <= V2(r) <= 0, then it follows that E1 <= E2. (II) If V(r,a) depends on a parameter a ∈(a1,a2), V(r,a) <= 0, and E(a) is any positive eigenvalue, then ∂ V/∂ a >= 0 ==> E'(a) >= 0 and ∂ V/∂ a <= 0 ==> E'(a) <= 0.
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