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 Volodymyr Marchenko, is derived by computing the Fourier transform of the scattering relation: K ( r , r ′ ) + g ( r , r ′ ) + ∫ r ∞ K ( r , r ′ ′ ) g ( r ′ ′ , r ′ ) d r ′ ′ = 0 {\displaystyle 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 ′ ) {\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.