Abstract
In conventional scattering theory, to obtain an explicit result, one imposes a precondition that the distance between target and observer is infinite. With the help of this precondition, one can asymptotically replace the Hankel function and the Bessel function with the sine functions so that one can achieve an explicit result. Nevertheless, after such a treatment, the information of the distance between target and observer is inevitably lost. In this paper, we show that such a precondition is not necessary: without losing any information of distance, one can still obtain an explicit result of a scattering rigorously. In other words, we give an rigorous explicit scattering result which contains the information of distance between target and observer. We show that at a finite distance, a modification factor --- the Bessel polynomial --- appears in the scattering amplitude, and, consequently, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase (the argument of the Bessel polynomial) appears in the scattering wave function.
Highlights
Speaking, the above two treatments in conventional theory are to replace the spherical Hankel function, h(l1) and h(l2), and the spherical Bessel function, jl, with their asymptotics, and, inevitably lead to the loss of information of the distance r
We show that at a finite distance, a modification factor — the Bessel polynomial — appears in the scattering amplitude, and, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase appears in the scattering wave function
Because the outgoing waves are different at different distances, there is no uniform expression of the asymptotic boundary condition like eq (1.2)
Summary
A rigorous treatment without large-distance asymptotics for short-range potentials is established. The scattering wave function, scattering amplitude, phase shift, cross section, and a description of the outgoing wave are rigorously obtained
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