Marchenko equation

In mathematical physics, the Marchenko equation is a solution of the inverse scattering problem, when the energy of the input particle is a variable, and the impact parameter b is constant.

The main equation is:

$K(r,r^\prime) + g(r,r^\prime) + \int_r^{\infty} K(r,r^{\prime\prime}) g(r^{\prime\prime},r^\prime) \mathrm{d}r^{\prime\prime} = 0$

where

$g(r,r^\prime)\,$

is a symmetric kernel, so that

$g(r,r^\prime)=g(r^\prime,r).\,$

This equation is derived from the Gel'fand-Levitan integral equation, using the Povzner-Levitan representation.