Sharp comparison theorems for the Klein–Gordon equation in d dimensions

Abstract

We establish sharp (or ’refined’) comparison theorems for the Klein–Gordon equation. We show that the condition [Formula: see text], which leads to [Formula: see text], can be replaced by the weaker assumption [Formula: see text] which still implies the spectral ordering [Formula: see text]. In the simplest case, for [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] and for [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. We also consider sharp comparison theorems in the presence of a scalar potential [Formula: see text] (a ‘variable mass’) in addition to the vector term [Formula: see text] (the time component of a four-vector). The theorems are illustrated by a variety of explicit detailed examples.