Eur. Phys. J. Plus (2016) 131: 102
https://doi.org/10.1140/epjp/i2016-16102-y

Regular Article

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

Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, H3G 1M8, Montréal, Québec, Canada

^* e-mail: richard.hall@concordia.ca

Received: 15 January 2016
Accepted: 14 February 2016
Published online: 18 April 2016

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 , then . 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 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 dimensions. For instance in a simplest case in one dimension, the condition is replaced by , where , , and or b.