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 $V_a\le V_b$, 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.

17 pages, 9 figures. The paper has been extensively re-written to improve the clarity and completeness of the presentation

References in corpus (4)

• Solutions to the 1d Klein-Gordon equation with cutoff Coulomb potentials
• Special comparison theorem for the Dirac equation
• Comparison theorems for the Klein-Gordon equation in d dimensions
• Comparison theorems for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions

Cited by in corpus (1)

• General comparison theorems for the Klein-Gordon equation in d dimensions