The overcritical Dirac–Coulomb operator

Abstract

Considering relativistic hydrogenic systems with one electron moving around a fixed nuclear centre of charge Ze0/α, we analyse the corresponding Dirac–Coulomb operator; our focus here lies on the Z-regime in which this operator is no longer essentially self-adjoint but allows various self-adjoint extensions. We characterize these self-adjoint extensions by appropriate boundary conditions at the nuclear centre and study their spectral as well as other relevant properties. A new phenomenon that can be described as ‘Rydberg doubling’ is encountered in the overcritical regime Z > 1, namely for all self-adjoint extensions of the Dirac–Coulomb Hamiltonian a second Rydberg series converging to the threshold of the negative continuous spectrum emerges. The occurrence of this effect also sheds some new light on the frequently discussed problem of the ‘charged vacuum’.