Method of Moments in the Collision Theory

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Abstract
The scattering matrix elements for two-body collision problems are obtained in terms of reactance matrix elements which are in turn given in terms of solutions of moment problems. The reactance matrix elements obtained by this method are suitable for practical calculations. Nonsingular as well as optical potentials are used in this study and illustrative calculations are given. The phase shifts calculated with an attractive exponential potential by means of the first moment of symmetrized integral kernel gives a reasonable agreement with the exact phase shifts for the parameter chosen. General forms and computational prescriptions for a semiclassical optical phase shift and wavefunction are given. For an optical potential with a δ function as its imaginary part the first moment gives rigorous result. Optical phase shifts for this potential are calculated as a function of orbital angular momentum at a given kinetic energy. It is found that the imaginary part of the phase shift has a series of sharp spikes, which may render an explanation of the undulation at large angles of the differential elastic cross section of reactive systems. It is also shown that the optical phase shift obtained in this method reduces in high-energy limit to a form similar to that in the eikonal approximation.