Zero-range potential for particles interacting via Coulomb potential

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 wavefunction 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 us 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 in the case of the imaginary absorbing potential in the presence of the Coulomb force is given in terms of the partial wave series. (paper)