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

摘要

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.