Analytical properties of the amplitude for elastic forward scattering of electrons by atoms

It is shown that the exchange amplitude for elastic forward scattering has an additional cut at the left semi-axis of the complex energy plane. The branching point is located at the binding energy of the electron in the atom. For scattering by hydrogen atoms the amplitude diverges logarithmically at E=-1 Ry and hence the branching point is logarithmic. On the left hand axis of the physical sheet no singularities beside the poles corresponding to bound states of the negative ion have been observed. Although branching points exist at energies equal to the ionization potentials of the atomic excited states, they lie in the nonphysical sheets and hence do not affect the dispersion relation. The position and nature of the singularity is completely determined dispersion relation. The position and nature of the singularity is completely determined by the shape of the interaction potential. It also exists for short-range potentials but is considerably removed from the physical region, whereas in the case of coulomb forces it is close to it and plays an essential role in the dispersion relation