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

where the $$\beta_n$$ are non-zero constants, solving the GLM equation

for $$K$$ allows the potential to be recovered using the formula