Comparison theorems for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions

TL;DR

This paper establishes comparison theorems for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions, enabling spectral approximations using known solutions and deriving bounds for specific potentials.

Contribution

It introduces comparison theorems for Dirac eigenvalues under symmetric interactions and applies envelope theory to generate spectral bounds using exact solutions.

Findings

Eigenvalues are ordered when potentials are ordered.

Envelope theory provides spectral approximations.

Analytical bounds are derived for the log potential.

Abstract

A single Dirac particle is bound in d dimensions by vector V(r) and scalar S(r) central potentials. The spin-symmetric S=V and pseudo-spin-symmetric S = - V cases are studied and it is shown that if two such potentials are ordered V^{(1)} \le V^{(2)}, then corresponding discrete eigenvalues are all similarly ordered E_{\kappa \nu}^{(1)} \le E_{\kappa \nu}^{(2)}. This comparison theorem allows us to use envelope theory to generate spectral approximations with the aid of known exact solutions, such as those for Coulombic, harmonic-oscillator, and Kratzer potentials. The example of the log potential V(r) = v\ln(r) is presented. Since $V(r)$ is a convex transformation of the soluble Coulomb potential, this leads to a compact analytical formula for lower-bounds to the discrete spectrum. The resulting ground-state lower-bound curve E_{L}(v) is compared with an accurate graph found by direct numerical integration.