Marchenko equation

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 $${\displaystyle u(x)=-2{\frac {d}{dx}}K(x,x).}$$

• Lax pair