Off-shell CoulombTmatrix in connection with the exact solution of three-particle equations with Coulomb interaction

We investigate the problems connected with the incorporation of the Coulomb interaction into the equations which describe reactions involving charged particles. In particular, the formalism of Vesselova and of Alt, Sandhas, and Ziegelmann is considered. Here various Coulomb quantities play a role which all have a so-called essential singularity in the zero-energy point. This singularity has to be taken care of when the total three-particle energy is near the breakup threshold. We use formulas, both for the case of a repulsive and an attractive Coulomb potential, by means of which the essential singularity is split off in a satisfactory way. This allows one to make really exact calculations for simple models. It is shown that a comparison of such exact calculations with known approximations is indeed necessary. We present and discuss exact calculations of the three-dimensional Coulomb $T$ matrix and of its partial wave projections for all real energies.