On dynamical justification of quantum scattering cross section

We suggest a dynamical justification of quantum differential cross section in the context of long-time transition to stationary regime for the Schrödinger equation. The problem has been stated by Reed and Simon. Our approach is based on spherical incident waves produced by a harmonic source and the long-range asymptotics for the corresponding spherical limiting amplitudes. The main results are as follows: i) the convergence of spherical limiting amplitudes to the limit as the source goes away to infinity, and ii) the proof of the coincidence of the corresponding limit scattering cross section with the universally recognized formula. The main technical ingredients are the Agmon-Jensen-Kato's analytical theory of the Green function, Ikebe's uniqueness theorem for the Lippmann-Schwinger equation, and some refinement of classical long-range asymptotics for the Coulomb potentials.