Physics: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:
- [math]\displaystyle{ 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]\displaystyle{ g(r,r^\prime)\, }[/math]is a symmetric kernel, such that [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ u(x) }[/math] for the Schrödinger operator [math]\displaystyle{ L = -\frac{d^2}{dx^2} + u(x) }[/math], one has the scattering data [math]\displaystyle{ (r(k), \{\chi_1, \cdots, \chi_N\}) }[/math], where [math]\displaystyle{ r(k) }[/math] are the reflection coefficients from continuous scattering, given as a function [math]\displaystyle{ r: \mathbb{R} \rightarrow \mathbb{C} }[/math], and the real parameters [math]\displaystyle{ \chi_1, \cdots, \chi_N \gt 0 }[/math] are from the discrete bound spectrum.[1]
Then defining [math]\displaystyle{ 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]\displaystyle{ \beta_n }[/math] are non-zero constants, solving the GLM equation [math]\displaystyle{ K(x,y) + F(x+y) + \int_x^\infty K(x,z) F(z+y) dz = 0 }[/math] for [math]\displaystyle{ K }[/math] allows the potential to be recovered using the formula [math]\displaystyle{ u(x) = -2 \frac{d}{dx}K(x,x). }[/math]
See also
References
- ↑ Dunajski, Maciej (2015). Solitons, instantons, and twistors (1. publ., corrected 2015 ed.). Oxford: Oxford University Press. ISBN 978-0198570639.
- Marchenko, V. A. (2011). Sturm–Liouville Operators and Applications (2nd ed.). Providence: American Mathematical Society. ISBN 978-0-8218-5316-0.
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