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.

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

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

See also

• Lax pair

References

• Marchenko, V. A. (2011). Sturm–Liouville Operators and Applications (2nd ed.). Providence: American Mathematical Society. ISBN 978-0-8218-5316-0. MR 2798059.