Scattering theory without large-distance asymptotics in arbitrary dimensions

Abstract

In conventional scattering theory, by large-distance asymptotics, at the cost of losing the information of the distance between target and observer, one imposes a large-distance asymptotics to achieve a scattering wave function which can be represented explicitly by a scattering phase shift. In this paper, without large-distance asymptotics, we establish an arbitrary-dimensional scattering theory. Arbitrary-dimensional scattering wave functions, scattering boundary conditions, cross sections, and phase shifts are given without large-distance asymptotics. The importance of an arbitrary-dimensional scattering theory is that the dimensional renormalization procedure in quantum field theory needs an arbitrary-dimensional result. Moreover, we give a discussion of one- and two-dimensional scatterings.