Gelfand–Levitan–Marchenko equation

Summary

The Gelfand–Levitan–Marchenko equation is a linear integral equation that recovers a potential $u$ in the one-dimensional Schrödinger operator $-\frac{d^2}{dx^2} + u(x)$ from its scattering data. The data are the reflection coefficient, the bound state energies and their norming constants; they enter the equation through a single function $F$. The solution $K(x, y)$ gives the potential by $u(x) = -2 \frac{d}{dx} K(x, x)$. It is the central step of the inverse scattering transform, which solves the Korteweg–De Vries equation and other integrable equations.