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.[1]

Then defining

F(x)=\sum _{n=1}^{N}\beta _{n}e^{-\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).

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. MR 2798059.

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[dunajski-1] Dunajski, Maciej (2015). Solitons, instantons, and twistors (1. publ., corrected 2015 ed.). Oxford: Oxford University Press. ISBN 978-0198570639.

Marchenko equation

Application to scattering theory

See also

References