Positron production in heavy-ion collisions. I. Analytical description of the resonance in the positron continuum for supercritical atoms

With increasing nuclear charge $Z$ (and finite nuclear size), the lowest bound-state solution of the Dirac equation changes into a resonance located in the positron continuum (supercritical atom). We develop a formalism which enables us to treat this resonance explicitly in the manner of a quasibound state. The processes of electron and positron production in heavy-ion collisions of total charge $Z_1+Z_2\gtrsim 170$ can then be formulated analytically in such a way that all the rapid energy dependence due to this resonance is made explicit. The definition of the resonance wave function involves some ambiguity. As a consequence, the decomposition of the transition amplitude for positron production into a "spontaneous" and an "induced" part cannot be given unambiguously. The coupled time-dependent equations for the occupation amplitudes of the adiabatic single-particle states contain matrix elements which are smooth functions of the energy, and which can be calculated easily. The application of this formalism to positron production in supercritical atoms is presented in the following article by one of us (T.T.).