Refined comparison theorems for the Dirac equation in d dimensions

TL;DR

This paper refines relativistic comparison theorems for the Dirac equation in multiple dimensions, allowing potential graphs to cross while still ensuring spectral ordering of eigenvalues, thus broadening the applicability of spectral bounds.

Contribution

The work introduces refined comparison theorems for the Dirac equation that permit potential crossover, extending previous results and providing more flexible spectral ordering conditions.

Findings

Theorems apply to 1D and higher dimensions with specific integral conditions.

Potential crossover is allowed under controlled integral inequalities.

Spectral ordering $E_a _b$ is maintained under these new conditions.

Abstract

A single spin-$\frac{1}{2}$ particle obeys the Dirac equation in $d\ge 1$ spatial dimension and is bound by an attractive central monotone potential which vanishes at infinity (in one dimension the potential is even). This work refines the relativistic comparison theorems which were derived by Hall \cite{p75}. The new theorems allow the graphs of the two comparison potentials $V_a$ and $V_b$ to crossover in a controlled way and still imply the spectral ordering $E_a\le E_b$ for the eigenvalues at the bottom of each angular momentum subspace. More specifically in a simplest case we have: in dimension $d=1$, if $\int_0^x (V_b(t)-V_a(t)) dt\ge 0,\ x\in [0,\ \infty)$, then $E_a\le E_b$; and in $d>1$ dimensions, if $\int_0^r (V_b(t)-V_a(t))t^{2|k_d|} dt\ge 0,\ r\in [0,\ \infty)$, where $k_d=\tau\left(j+\frac{d-2}{2}\right)$ and $\tau=\pm 1$, then $E_a\le E_b$.