Marchenko equation

In mathematical physics, more specific in the one-dimensional inverse scattering problem, the Marchenko equation, named after Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation:

where

is a symmetric kernel, so that

which is computed from the scattering data. Solving the Marchenko equation one obtains the kernel of the transformation operator K ( r , r ′ ) from which the potential can be read off.

This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation.