Marchenko equation

In mathematical physics, more specifically the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation: In mathematical physics, more specifically the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation: Where g ( r , r ′ ) {displaystyle g(r,r^{prime }),} is a symmetric kernel, such that g ( r , r ′ ) = g ( r ′ , r ) , {displaystyle g(r,r^{prime })=g(r^{prime },r),,} which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator K ( r , r ′ ) {displaystyle K(r,r^{prime })} from which the potential can be read off. This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation.