Marchenko equation explained

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 ^{inf ty} _{r} K(r,r ^{\prime\prime} ) g(r ^{\prime\prime} ,r ^{\prime)} 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 =- (d ^{2})/(dx ^{2}) + u(x) , one has the scattering data (r(k), \{\chi _{1,} … , \chi _{N\})} , where r(k) are the reflection coefficients from continuous scattering, given as a function r: R → C , and the real parameters \chi _{1,} … , \chi _{N} > 0 are from the discrete bound spectrum.

Then defining $$F(x)\; =\; \backslash sum\_^N\backslash beta\_ne^\; +\; \backslash frac\; \backslash int\_\backslash mathbbr(k)e^dk,$$where the

\beta _{n} are non-zero constants, solving the GLM equation$$K(x,y)\; +\; F(x+y)\; +\; \backslash int\_x^\backslash infty\; K(x,z)\; F(z+y)\; dz\; =\; 0$$for K allows the potential to be recovered using the formula$$u(x)\; =\; -2\; \backslash fracK(x,x).$$

See also

• Lax pair

References

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