The relativistic mean-field equations of the atomic nucleus

TL;DR

This paper proves the existence of solutions to the relativistic mean-field equations in nuclear physics by employing a minimization approach with spectral constraints and the concentration-compactness lemma.

Contribution

It introduces a rigorous mathematical proof of solutions to the relativistic mean-field equations without nonlinear sigma meson self-coupling.

Findings

Existence of solutions is established mathematically.

A minimization framework with spectral constraints is developed.

The concentration-compactness lemma is applied successfully.

Abstract

In nuclear physics, the relativistic mean-field theory describes the nucleus as a system of Dirac nucleons which interact via meson fields. In a static case and without nonlinear self-coupling of the $\sigma$ meson, the relativistic mean-field equations become a system of Dirac equations where the potential is given by the meson and photon fields. The aim of this work is to prove the existence of solutions of these equations. We consider a minimization problem with constraints that involve negative spectral projectors and we apply the concentration-compactness lemma to find a minimizer of this problem. We show that this minimizer is a solution of the relativistic mean-field equations considered.