Coulomb problem for a $ Z>Z_{\rm cr}$ nucleus

Abstract

A closed-form equation is derived for the critical nucleus charge at which a discrete level with the Dirac quantum number touches the lower continuum of the Dirac equation solutions. For the Coulomb potential cut off rectangularly at the short distance , , the critical nucleus charge values are obtained for several values of and . It is shown that the partial scattering matrix of elastic positron-nucleus scattering, , is also unitary for . For this range, the scattering phase is calculated as a function of the positron energy = , as are the positions and widths of quasidiscrete levels corresponding to the scattering matrix poles. The implication is that the single-particle approximation for the Dirac equation is valid not only for but also for and that there is no spontaneous creation of pairs from the vacuum.