Refining the general comparison theorem for Klein-Gordon equation

By recasting the Klein–Gordon equation as an eigen-equation in the coupling parameter $v>0,$ the basic Klein–Gordon comparison theorem may be written $f_1\le f_2⟹G_1(E)\le G_2(E)$, where $f_1$ and $f_2$, are the monotone non-decreasing shapes of two central potentials $V_1(r)=v_1{f}_1(r)$ and $V_2(r)=v_2{f}_2(r)$ on $[0,\mathrm{\infty })$. Meanwhile $v_1=G_1(E)$ and $v_2 = G_2(E)$ are the corresponding coupling parameters that are functions of the energy $E\in (-m,m)$. We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in $d=1$ dimension) that if ${\int }_0^x[f_2(t)-f_1(t)]{\phi }_i(t)dt\ge 0$, the couplings remain ordered $v_1\le v_2$ where $i=1\mathrm{o}\mathrm{r}2,$ and ${\varphi_1, \varphi_2}$aretheground-statescorrespondingrespectivelytothecouplings${v_1,, v_2}$foragiven$E \in (-m,, m).$. This result is extended to spherically symmetric radial potentials in $d>1$ dimensions.