Complex-energy solutions of the relativistic Kepler problem at high energies

Abstract

In the relativistic Kepler problem the analytic continuation of the principal quantum number $n$ into $i\ensuremath{\nu}$ describes the scattering states. When ${\ensuremath{\nu}}^{2}$ ranges from $+\ensuremath{\infty}$ to ${\ensuremath{\alpha}}^{2}$ the energy ranges from (${m}_{1}+{m}_{2}$) to infinity. In the remaining gap $0<{\ensuremath{\nu}}^{2}<{\ensuremath{\alpha}}^{2}$, the energy becomes complex corresponding to a continuum of resonances or perhaps pair production. This phenomenon also exists in the classical relativistic problem, and we suggest that one should look experimentally for the manifestation of this gap, for example when $Z>~137$.