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.

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