Mathematics

Central in relativistic quantum mechanics is the Dirac equation, a partial differential equation that describes the dynamics of a spin-1/2 fermion. When one studies such dynamics in the electrostatic field generated by a nuclear point charge in the origin, one has to face the problem of qualifying the self-adjoint realisation(s) of the minimal Dirac-Coulomb operator. For sufficiently large atomic number, it is well known that there is indeed an infinite multiplicity of physically distinct realisations. We will survey the state-of-art of the problem and the main open questions, together with an explicit characterisation of the Dirac-Coulomb self-adjoint extensions in the so-called sub-critical regime, recently obtained in a joint work with Alessandro Michelangeli.