The zero range potential is constructed for a system of two particles interacting via the Coulomb potential. The singular part of the asymptote of the wave function at the origin which is caused by the common effect of the zero range potential singularity and of the Coulomb potential is explicitly calculated by using the Lippmann-Schwinger type integral equation. The singular pseudo potential is constructed from the requirement that it enforces the solution to the Coulomb Schrödinger equation to possess the calculated asymptotic behavior at the origin. This pseudo potential is then used for constructing a model of the imaginary absorbing potential for the positron electron system. This potential allows to treat the annihilation process in positron electron collisions on the basis of the non relativistic Schrödinger equation. The functional form of the pseudo potential constructed in this paper is analogous to the well known Fermi-Breit-Huang pseudo potential. The generalization of the optical theorem on the case of the imaginary absorbing potential in presence of the Coulomb force is given in terms of the partial wave series.

The potential splitting approach incorporated into the framework of Faddeev-Merkuriev equations in the differential form is used for calculations of multichannel scattering in e − e +p and e + e − He ++ systems. Detailed calculations of all possible Swave cross-sections are performed in the low-energy region which supports up to seven open channels including the rearrangement channels of ground and excited states of antihydrogen, positronium and helium ion formations. All known sharp resonances of the systems obtained and approved by a number of authors are clearly reproduced in the calculated cross sections. In cross sections for energies above the threshold corresponding to n = 2 state of antihydrogen the prominent oscillations of Gailitis Damburg type have been found.

We present the results of positron-Hydrogen multichannel scattering calculations performed on the base of Faddeev-Merkuriev equations. We discuss an optimal choice of the Merkuriev's Coulomb splitting parameters. Splitting the Coulomb potential in two-body configuration space is applicable for a limited energy range. Splitting the potential in three-body configuration space makes it possible to perform calculations in a broader range of energies and to optimize the numerical convergence. Scattering cross sections for zero total angular momentum for all processes between the positronium formation threshold and the third excitation threshold of the Hydrogen atom are reported.

The computational approach for solving the Faddeev-Merkuriev equations in total orbital momentum representation is presented. These equations describe a system of three quantum charged particles and are widely used in bound state and scattering calculations. The approach is based on the spline collocation method and exploits intensively the tensor product form of discretized operators and preconditioner, which leads to a drastic economy in both computer resources and time.

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