{{Short description|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 Volodymyr Marchenko, is derived by computing the 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, 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, using the Povzner–Levitan representation.
== Application to scattering theory == Suppose that for a potential <math>u(x)</math> for the Schrödinger operator <math>L = -\frac{d^2}{dx^2} + u(x)</math>, one has the 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
==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 |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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