Scattering theory without large-distance asymptotics

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.

Exact solution of inverse-square-root potential V(r)=−αr

“…The inverse-power potential V (r) ∼ 1/r s with 0 < s < 2 is a long-range potential and has both scattering states and bound states; while V (r) ∼ 1/r s with s ≥ 2 is a shortrange potential and has only scattering states. The short-range power potential, 1/r s with s ≥ 2, has only scattering states and can be generally treated [1,2]. The long-range power potential, 1/r s with 0 < s < 2, only when s = 1, the Coulomb potential, is exactly solved.…”

Scalar scattering in Schwarzschild spacetime: Integral equation method

“…It is worthy to note that scattering by Schwarzschild spacetime is a long-range potential scattering [1], i.e., this is an integral equation method for long-range potential scattering. The long-range scattering is more difficult than short-range scattering [31,[34][35][36].…”

Scattering state and bound state of scalar field in Schwarzschild spacetime: Exact solution

“…Furthermore, the boundary condition at r → ∞ is the asymptotic solution of the radial equation (2.2) at r → ∞ [3,[39][40][41]. The asymptotics of the radial equation (2.2) at r → ∞ is 1…”

“…Before seeking exact solutions, we first investigate the large-distance asymptotic behavior of the scattering wave function. The asymptotic solution of the scattering wave function will serve as boundary conditions for scattering states [41,43,44].…”