  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.
