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.

Application to scattering theory

Suppose that for a potential $u(x)$ for the Schrödinger operator $L = -\frac + u(x)$, one has the scattering data $(r(k), \)$, where $r(k)$ are the reflection coefficients from continuous scattering, given as a function $r: \mathbb \rightarrow \mathbb$, and the real parameters $\chi_1, \cdots, \chi_N > 0$ are from the discrete bound spectrum.

Then defining
$F(x) = \sum_^N\beta_ne^ + \frac \int_\mathbbr(k)e^dk,$
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
$u(x) = -2 \fracK(x,x).$

See also

* Lax pair

Notes

References

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Eponymous equations of physics
Integral equations
Scattering theory

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