标题： 散射理论无需远距离渐近分析 显示英文标题

标题： Scattering theory without large-distance asymptotics

摘要： 在传统散射理论中，为了获得一个明确的结果，需要施加一个前提条件，即目标与观察者之间的距离是无限的。 借助这个前提条件，可以渐近地用正弦函数替换汉克尔函数和贝塞尔函数，从而得到一个明确的结果。 然而，经过这样的处理后，目标与观察者之间的距离信息不可避免地丢失了。 在本文中，我们表明这个前提条件是不必要的：在不丢失任何距离信息的情况下，仍然可以严格地获得一个明确的散射结果。 换句话说，我们给出了一个包含目标与观察者之间距离信息的严格明确的散射结果。 我们表明，在有限距离下，散射振幅中会出现一个修正因子——贝塞尔多项式，因此截面依赖于距离，出射波前表面不再是球面，并且除了相位移外，散射波函数中还会出现一个额外的相位（贝塞尔多项式的幅角）。 显示英文摘要

摘要： 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.