# Marchenko equation

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{{Short description|Integral equation}}
In [mathematical physics](/source/mathematical_physics), more specifically the one-dimensional [inverse scattering problem](/source/inverse_scattering_problem), the '''Marchenko equation''' (or '''Gelfand-Levitan-Marchenko equation''' or '''GLM equation'''), named after [Israel Gelfand](/source/Israel_Gelfand), [Boris Levitan](/source/Boris_Levitan) and [Volodymyr Marchenko](/source/Volodymyr_Marchenko), is derived by computing the [Fourier transform](/source/Fourier_transform) of the scattering relation:
:<math>
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
</math>

Where <math>g(r,r^\prime)\,</math>is a [symmetric kernel](/source/symmetric_kernel), such that <math>g(r,r^\prime)=g(r^\prime,r),\,</math>which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator <math>K(r,r^\prime)</math> from which the potential can be read off. This equation is derived from the [Gelfand–Levitan integral equation](/source/Gelfand%E2%80%93Levitan_integral_equation), using the [Povzner–Levitan representation](/source/Povzner%E2%80%93Levitan_representation). 

== Application to scattering theory ==
Suppose that for a potential <math>u(x)</math> for the [Schrödinger operator](/source/Schr%C3%B6dinger_operator) <math>L = -\frac{d^2}{dx^2} + u(x)</math>, one has the [scattering](/source/scattering) data <math>(r(k), \{\chi_1, \cdots, \chi_N\})</math>, where <math>r(k)</math> are the reflection coefficients from continuous scattering, given as a function <math>r: \mathbb{R} \rightarrow \mathbb{C}</math>, and the real parameters <math>\chi_1, \cdots, \chi_N > 0</math> are from the discrete bound spectrum.{{sfn | Dunajski | 2009 | pp=30-31}}

Then defining 
<math display = block>F(x) = \sum_{n=1}^N\beta_ne^{-\chi_n x} + \frac{1}{2\pi} \int_\mathbb{R}r(k)e^{ikx}dk,</math>
where the <math>\beta_n</math> are non-zero constants, solving the GLM equation
<math display = block>K(x,y) + F(x+y) + \int_x^\infty K(x,z) F(z+y) dz = 0</math>
for <math>K</math> allows the potential to be recovered using the formula
<math display = block> u(x) = -2 \frac{d}{dx}K(x,x).</math>

== See also ==

* [Lax pair](/source/Lax_pair)

==Notes==
{{reflist}}

==References==
* {{cite book | last=Dunajski | first=Maciej | title=Solitons, Instantons, and Twistors | publisher=OUP Oxford | publication-place=Oxford; New York | year=2009 | isbn=978-0-19-857063-9 | oclc=320199531}}
* {{cite book |mr=2798059 |last1=Marchenko |first1=V. A. |title=Sturm–Liouville Operators and Applications |edition=2nd |publisher=[American Mathematical Society](/source/American_Mathematical_Society) |location=Providence |year=2011 |isbn=978-0-8218-5316-0 }}
* {{cite book | last=Kay | first=Irvin W. | title=The inverse scattering problem | publisher=Courant Institute of Mathematical Sciences, New York University | publication-place=New York | year=1955 | oclc=1046812324 |url=https://archive.org/details/inversescatterin00kayi/page/n3/mode/2up}}
* {{cite journal | last=Levinson | first=Norman | title=Certain Explicit Relationships between Phase Shift and Scattering Potential | journal=Physical Review | volume=89 | issue=4 | year=1953 | issn=0031-899X | doi=10.1103/PhysRev.89.755 | pages=755–757| bibcode=1953PhRv...89..755L }}

{{DEFAULTSORT:Marchenko Equation}}
Category:Integral equations
Category:Scattering theory

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Adapted from the Wikipedia article [Marchenko equation](https://en.wikipedia.org/wiki/Marchenko_equation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Marchenko_equation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.