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:

$K(r,r^{\prime })+g(r,r^{\prime })+\underset{r}{\overset{\mathrm{\infty }}{\int }}K(r,r^{\prime \prime })g(r^{\prime \prime },r^{\prime })\mathrm{d}{r}^{\prime \prime }=0$

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 $L=-\frac{d^2}{d{x}^2}+u(x)$, 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 $r:ℝ\to ℂ$, and the real parameters ${\chi }_1,\cdots ,{\chi }_N>0$ are from the discrete bound spectrum.Template:Sfn

Then defining $$F(x)=\sum _{n=1}^N{\beta }_n{e}^{-{\chi }_{n}x}+\frac{1}{2\pi }\underset{ℝ}{\int }r(k)e^{ikx}dk,$$ where the ${\beta }_n$ are non-zero constants, solving the GLM equation $$K(x,y)+F(x+y)+\underset{x}{\overset{\mathrm{\infty }}{\int }}K(x,z)F(z+y)dz=0$$ 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

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite journal

Template:Scattering-stub