Sharp comparison theorems for the Klein--Gordon equation in $d$ dimensions

TL;DR

This paper develops refined comparison theorems for the Klein-Gordon equation, allowing spectral ordering to be established under weaker conditions involving integrated potentials, with applications to variable mass scenarios.

Contribution

It introduces sharper comparison theorems for the Klein-Gordon equation that replace pointwise potential inequalities with integrated conditions, extending to scalar potentials.

Findings

Established that $U_a \\le U_b$ implies $E_a \\le E_b$ under weaker conditions.

Derived comparison theorems for Klein-Gordon with scalar and vector potentials.

Provided explicit examples illustrating the theorems.

Abstract

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). The theorems are illustrated by a variety of explicit detailed examples.