Marchenko equation

In mathematical physics, more specific in 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:

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

where

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

is a symmetric kernel, so that

$g(r,r^{\prime })=g(r^{\prime },r),\,$

which is computed from the scattering data. Solving the Marchenko equation one obtains the kernel of the transformation operator $K(r,r^{\prime })$ from which the potential can be read off.

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

References

• Marchenko, V. A. (2011). Sturm–Liouville Operators and Applications (2nd ed.). Providence: American Mathematical Society. ISBN 978-0-8218-5316-0. MR 2798059.