Refining the general comparison theorem for Klein-Gordon equation

TL;DR

This paper refines the comparison theorem for the Klein-Gordon equation by weakening conditions for spectral ordering and extending results to higher dimensions with spherically symmetric potentials.

Contribution

It introduces a less restrictive condition for spectral ordering and extends the comparison theorem to multi-dimensional spherically symmetric potentials.

Findings

Weaker integral condition ensures spectral ordering for ground states.

Extension of the comparison theorem to higher dimensions with spherical symmetry.

Applicable to potentials with monotone non-decreasing shapes.

Abstract

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 $f_1\leq f_2\implies G_1(E)\leq G_2(E)$, 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 $[0,\infty)$. 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 $\int_0^x\big[f_2(t) - f_1(t)\big]\varphi_i(t)dt\geq 0$, the couplings remain ordered $v_1 \leq v_2$ where $i = 1\, {\rm or}\, 2, $ and $\{\varphi_1, \varphi_2\}$ are the ground-states corresponding respectively to the couplings $\{v_1,\, v_2\}$ for a given $E \in (-m,\, m).$. This result is extended to spherically symmetric radial potentials in $ d > 1 $ dimensions.