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:

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

Where ${\displaystyle g(r,r^{\prime })\,}$is a symmetric kernel, such that ${\displaystyle g(r,r^{\prime })=g(r^{\prime },r),\,}$which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator ${\displaystyle K(r,r^{\prime })}$ from which the potential can be read off. This equation is derived from the Gelfand–Levitan integral equation, using the Povzner–Levitan representation.

Application to scattering theory

Suppose that for a potential ${\displaystyle u(x)}$ for the Schrödinger operator ${\displaystyle L=-{\frac {d^{2)){dx^{2))}+u(x)}$, one has the scattering data ${\displaystyle (r(k),\{\chi _{1},\cdots ,\chi _{N}\})}$, where ${\displaystyle r(k)}$ are the reflection coefficients from continuous scattering, given as a function ${\displaystyle r:\mathbb {R} \rightarrow \mathbb {C} }$, and the real parameters ${\displaystyle \chi _{1},\cdots ,\chi _{N}>0}$ are from the discrete bound spectrum.^[1]

Then defining

$${\displaystyle F(x)=\sum _{n=1}^{N}\beta _{n}e^{-\chi _{n}x}+{\frac {1}{2\pi ))\int _{\mathbb {R} }r(k)e^{ikx}dk,}$$

${\displaystyle \beta _{n))$

$${\displaystyle K(x,y)+F(x+y)+\int _{x}^{\infty }K(x,z)F(z+y)dz=0}$$

${\displaystyle K}$

$${\displaystyle u(x)=-2{\frac {d}{dx))K(x,x).}$$