Scattering theory without large-distance asymptotics: scattering boundary condition

TL;DR

This paper develops a rigorous scattering theory that does not rely on large-distance asymptotics, preserving distance information and defining a scattering amplitude dependent only on the scattering angle.

Contribution

It introduces a new scattering boundary condition and a distance-independent scattering amplitude, extending conventional theory without losing distance information.

Findings

Explicit scattering wave function with phase shift preserving distance info

Scattering amplitude depending only on scattering angle

Recovery of conventional scattering amplitude in large-distance limit

Abstract

By large-distance asymptotics, in conventional scattering theory, at the cost of losing the information of the distance between target and observer, one arrives at an explicit expression for scattering wave functions represented by a scattering phase shift. In the present paper, together with a preceding paper (T. Liu,W.-D. Li, and W.-S. Dai, JHEP06(2014)087), we establish a rigorous scattering theory without imposing large-distance asymptotics. We show that even without large-distance asymptotics, one can also obtain an explicit scattering wave function represented also by a scattering phase shift, in which, of course, the information of the distance is preserved. Nevertheless, the scattering amplitude obtained in the preceding paper depends not only on the scattering angle but also on the distance between target and observer. In this paper, by constructing a scattering boundary condition without large-distance asymptotics, we introduce a scattering amplitude, like that in conventional scattering theory, depending only on the scattering angle and being independent of the distance. Such a scattering amplitude, when taking large-distance asymptotics, will recover the scattering amplitude in conventional scattering theory. The present paper, with the preceding paper, provides a complete scattering theory without large-distance asymptotics.