Relativistic comparison theorems for the Klein-Gordon equation with scalar and vector potentials in d-dimensions

Relativistic comparison theorems are established for discrete eigenvalues of Klein-Gordon equation with vector and scalar potentials in d-dimensions. Theorem 1: If V(λ) and S(λ) depend on a parameter λ, ∂S/∂λ?0, S?0, ∂V/∂λ?0, V?0, E>0, then it follows that ∂En/∂λ?0. Theorem 2: If S₂?S₁?0, 0?V₂?V₁, E>0, then the corresponding eigenvalues are ordered as En(2)?En(1). Theorem 3: If 0?V₂?V₁, S₂?|S₁|, En(1)>0, En(2)>0, then En(2)?En(1). Some illustrative examples are given.