The impact of sharp screening on the Coulomb scattering problem in three dimensions

The scattering problem for two particles interacting via theCoulomb potential isexamined for the case where the potential has a sharp cut-off at some distance.The problem is solved for two complementary situations, firstly, when theinterior part of the Coulomb potential is left in the Hamiltonian and, secondly,when the long-range tail is considered as the potential. The partial wave resultsare summed up to obtain the wavefunction in three dimensions. It is shownthat in the domains where the wavefunction is expected to be proportionalto the known solutions, the proportionality is given by an operator actingon the angular part of the wavefunction. The explicit representation for thisoperator is obtained in the basis of Legendre polynomials. We proposed adriven Schr¨odinger equation including an inhomogeneous term of the finiterange with purely outgoing asymptotics for its solution in the case of thethree-dimensional scattering problem with long-range potentials