A closed form equation is derived for the critical nucleus charge, i.e. for the value Z=Z_cr at which the discrete level with the Dirac quantum number touches the lower continuum of Dirac equation solutions. For a Coulomb potential cut off rectangularly at a small distance r₀ = Rħ/(mc), R ⪡ 1, the critical nucleus charge values are obtained for several values of κ and R. It is shown that the partial scattering matrix of the elastic positron-nucleus scattering, S_κ = exp (2iδ_κ(ε_p)), is also unitary for Z>Z_cr. For this range, the scattering phase δ_κ(ε_p) is calculated as a function of the positron energy E_p = ε_pmc², 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 ZZ_cr and that there is no spontaneous creation of e⁺e⁻ pairs from vacuum.

Keywords: the Coulomb problem, pointlike nucleus, boundary conditions, critical charge, scattering phase, Breit—Wigner resonance PACS: 03.65.Pm, 12.20.−m, 73.22.Pr (all) DOI: 10.3367/UFNe.0185.201508d.0845 URL: https://ufn.ru/en/articles/2015/8/d/ 000364717300004 2-s2.0-84947746838 2015PhyU...58..785K Citation: Kuleshov V M, Mur V D, Narozhnyi N B, Fedotov A M, Lozovik Yu E, Popov V S "Coulomb problem for a Z>Zcr" Phys. Usp. 58 785–791 (2015) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Received: 16th, June 2015, accepted: 23rd, June 2015 Оригинал: Кулешов В М, Мур В Д, Нарожный Н Б, Федотов А М, Лозовик Ю Е, Попов В С «Кулоновская задача с зарядом ядра Z>Zcr» УФН 185 845–852 (2015); DOI: 10.3367/UFNr.0185.201508d.0845