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In the simplest case, for $d=1$, $U_i(x)=\\int_0^x V_i(t)dt$, $i=a$ or $b$, and for $d>1$, $U_i(r)=\\int_0^r V_i(t) t^{d-1}dt$, $i=a$ or $b$. We also consider sharp comparison theorems in the presence of a scalar potential $S$ (a `variable mass') in addition to the vector term $V$ (the time component of a $4$-vector). 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Hall","submitted_at":"2015-06-04T21:05:14Z","abstract_excerpt":"We establish sharp (or `refined') comparison theorems for the Klein--Gordon equation. We show that the condition $V_a\\le V_b$, which leads to $E_a\\le E_b$, can be replaced by the weaker assumption $U_a\\le U_b$ which still implies the spectral ordering $E_a\\le E_b$. In the simplest case, for $d=1$, $U_i(x)=\\int_0^x V_i(t)dt$, $i=a$ or $b$, and for $d>1$, $U_i(r)=\\int_0^r V_i(t) t^{d-1}dt$, $i=a$ or $b$. We also consider sharp comparison theorems in the presence of a scalar potential $S$ (a `variable mass') in addition to the vector term $V$ (the time component of a $4$-vector). 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