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Marchenko 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^{\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, 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.

Application to scattering theory

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

Then defining

F(x) = \sum_{n=1}^N\beta_ne^{-\chi_n x} + \frac{1}{2\pi} \int_\mathbb{R}r(k)e^{ikx}dk,

where the \beta_n are non-zero constants, solving the GLM equation

K(x,y) + F(x+y) + \int_x^\infty K(x,z) F(z+y) dz = 0

for K allows the potential to be recovered using the formula

u(x) = -2 \frac{d}{dx}K(x,x).

See also

Notes

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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 }}

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Category:Eponymous equations of physics

Category:Integral equations

Category:Scattering theory

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