Refined comparison theorems for the Dirac equation with spin and pseudo--spin symmetry in $d$ dimensions

TL;DR

This paper enhances the comparison theorems for the Dirac equation with spin and pseudo-spin symmetry in multiple dimensions, allowing for crossing potentials and providing conditions under which energy ordering is preserved.

Contribution

It introduces refined comparison theorems that accommodate crossing potentials, extending applicability to ground states and angular momentum subspaces in relativistic quantum mechanics.

Findings

Comparison potentials can cross under certain conditions without losing energy ordering.

Theorems are valid for ground states in 1D and for the lowest angular momentum states in higher dimensions.

A new integral-based condition replaces pointwise potential ordering in 1D cases.

Abstract

The classic comparison theorem of quantum mechanics states that if two potentials are ordered then the corresponding energy eigenvalues are similarly ordered, that is to say if $V_a\le V_b$, then $E_a\le E_b$. Such theorems have recently been established for relativistic problems even though the discrete spectra are not easily characterized variationally. In this paper we improve on the basic comparison theorem for the Dirac equation with spin and pseudo--spin symmetry in $d\ge 1$ dimensions. The graphs of two comparison potentials may now cross each other in a prescribed manner implying that the energy values are still ordered. The refined comparison theorems are valid for the ground state in one dimension and for the bottom of an angular momentum subspace in $d>1$ dimensions. For instance in a simplest case in one dimension, the condition $V_a\le V_b$ is replaced by $U_a\le U_b$, where $U_i(x)=\int_0^x V_i(t)dt$, $x\in[0,\ \infty)$, and $i=a$ or $b$.