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 ′ ) + g ( r , r ′ ) + ∫ r ∞ K ( r , r ′ ′ ) g ( r ′ ′ , r ′ ) d r ′ ′ = 0 {\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 g ( r , r ′ ) {\displaystyle g(r,r^{\prime })\,} is a symmetric kernel, such that g ( r , r ′ ) = g ( r ′ , r ) , {\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 K ( r , r ′ ) {\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 u ( x ) {\displaystyle u(x)} for the Schrödinger operator L = − d 2 d x 2 + u ( x ) {\displaystyle L=-{\frac {d^{2}}{dx^{2}}}+u(x)} , one has the scattering data ( r ( k ) , { χ 1 , ⋯ , χ N } ) {\displaystyle (r(k),\{\chi _{1},\cdots ,\chi _{N}\})} , where r ( k ) {\displaystyle r(k)} are the reflection coefficients from continuous scattering, given as a function r : R → C {\displaystyle r:\mathbb {R} \rightarrow \mathbb {C} } , and the real parameters χ 1 , ⋯ , χ N > 0 {\displaystyle \chi _{1},\cdots ,\chi _{N}>0} are from the discrete bound spectrum.[1]

Then defining

F ( x ) = ∑ n = 1 N β n e − χ n x + 1 2 π ∫ R r ( k ) e i k x d k , {\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 β n {\displaystyle \beta _{n}} are non-zero constants, solving the GLM equation
K ( x , y ) + F ( x + y ) + ∫ x ∞ K ( x , z ) F ( z + y ) d z = 0 {\displaystyle K(x,y)+F(x+y)+\int _{x}^{\infty }K(x,z)F(z+y)dz=0}
for K {\displaystyle K} allows the potential to be recovered using the formula
u ( x ) = − 2 d d x K ( x , x ) . {\displaystyle u(x)=-2{\frac {d}{dx}}K(x,x).}

See also

  • Lax pair

References

  1. ^ 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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