Gelfand–Levitan–Marchenko Integral 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:
:$K(r,r^\prime) + g(r,r^\prime) + \int_r^ K(r,r^) g(r^,r^\prime) \mathrmr^ = 0$

Where $g(r,r^\prime)\,$is a symmetric kernel, such 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.

See also

* Lax pair

References

*

Integral equations
Scattering theory

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