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:

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

Where ${\displaystyle g(r,r^{\prime })\,}$is a symmetric kernel, such that ${\displaystyle g(r,r^{\prime })=g(r^{\prime },r),\,}$which is computed from the scattering data. Solving the Marchenko equation, one obtains the kernel of the transformation operator ${\displaystyle K(r,r^{\prime })}$ 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 ${\displaystyle u(x)}$ for the Schrödinger operator ${\displaystyle L=-{\frac {d^{2}}{dx^{2}}}+u(x)}$, one has the scattering data ${\displaystyle (r(k),\{\chi _{1},\cdots ,\chi _{N}\})}$, where ${\displaystyle r(k)}$ are the reflection coefficients from continuous scattering, given as a function ${\displaystyle r:\mathbb {R} \rightarrow \mathbb {C} }$, and the real parameters ${\displaystyle \chi _{1},\cdots ,\chi _{N}>0}$ are from the discrete bound spectrum.^[1]

Then defining $${\displaystyle F(x)=\sum _{n=1}^{N}\beta _{n}e^{-\chi _{n}x}+{\frac {1}{2\pi }}\int _{\mathbb {R} }r(k)e^{ikx}dk,}$$ where the ${\displaystyle \beta _{n}}$ are non-zero constants, solving the GLM equation $${\displaystyle K(x,y)+F(x+y)+\int _{x}^{\infty }K(x,z)F(z+y)dz=0}$$ for ${\displaystyle K}$ allows the potential to be recovered using the formula $${\displaystyle u(x)=-2{\frac {d}{dx}}K(x,x).}$$

See also

• Lax pair

Notes

1. Dunajski 2009, pp. 30–31.

References

• Dunajski, Maciej (2009). Solitons, Instantons, and Twistors. Oxford ; New York: OUP Oxford. ISBN 978-0-19-857063-9. OCLC 320199531.
• Marchenko, V. A. (2011). Sturm–Liouville Operators and Applications (2nd ed.). Providence: American Mathematical Society. ISBN 978-0-8218-5316-0. MR 2798059.
• Kay, Irvin W. (1955). The inverse scattering problem. New York: Courant Institute of Mathematical Sciences, New York University. OCLC 1046812324.
• Levinson, Norman (1953). "Certain Explicit Relationships between Phase Shift and Scattering Potential". Physical Review. 89 (4): 755–757. Bibcode:1953PhRv...89..755L. doi:10.1103/PhysRev.89.755. ISSN 0031-899X.