Qualitative Properties of the Dirac Equation in a Central Potential

TL;DR

This paper investigates the mathematical properties of the Dirac equation with a central potential, focusing on self-adjointness and spectral properties, especially for Coulomb-type potentials, using advanced operator theory methods.

Contribution

It provides a rigorous analysis of the self-adjointness conditions for the Dirac operator with Coulomb potential, including cases with anomalous magnetic moments, without assuming a fixed potential form.

Findings

Coulomb potential leads to non-infinitesimally bounded terms in the second-order equations.

Conditions for essential self-adjointness depend on the Coulomb parameter and angular momentum.

Methods extend to cases with non-negligible anomalous magnetic moments.

Abstract

The Dirac equation for a massive spin-1/2 field in a central potential V in three dimensions is studied without fixing a priori the functional form of V. The second-order equations for the radial parts of the spinor wave function are shown to involve a squared Dirac operator for the free case, whose essential self-adjointness is proved by using the Weyl limit point-limit circle criterion, and a `perturbation' resulting from the potential. One then finds that a potential of Coulomb type in the Dirac equation leads to a potential term in the above second-order equations which is not even infinitesimally form-bounded with respect to the free operator. Moreover, the conditions ensuring essential self-adjointness of the second-order operators in the interacting case are changed with respect to the free case, i.e. they are expressed by a majorization involving the parameter in the Coulomb potential and the angular momentum quantum number. The same methods are applied to the analysis of coupled eigenvalue equations when the anomalous magnetic moment of the electron is not neglected.