Analytic properties of the amplitude for the scattering of a particle by a central potential

Abstract

A new proof is given that the scattering amplitude, considered as a function of energy, for fixed momentum transfer, has those analytic properties in the energy variable which imply a dispersion relation. The proof is based on the known behavior of the Green's function, as the resolvent of a self-adjoint transformation, and on the study of the Born series. The class of potentials considered is the same as in previous studies. It is shown that for potentials of arbitrary strength the Born series converges uniformly to its first term for sufficiently high energies, real or complex. The analytic properties of the scattering amplitude as a function of momentum transfer for fixed real energy are also investigated and used to establish the domain of convergence of partial wave and related expansions. In particular the use of such expansions to define the amplitude in the unphysical region which occurs in the dispersion relation is fully justified for the same domain of momentum transfer for which the relation itself is valid, i.e.: Δ < 2α, where Δ is the momentum transfer andα −1 the range of the potential. All proofs apply equally to the Schrödinger and to the Klein-Gordon equations.