Gelfand–Levitan theory

Summary

Gelfand–Levitan theory reconstructs the potential $q$ of a Sturm–Liouville operator $-y'' + q y$ from its spectral measure. The key step is a linear integral equation, the Gelfand–Levitan equation, whose kernel is built from the spectral data. Its solution $K(x, t)$ is a transformation kernel that turns cosines into the solutions of the perturbed equation, and $q(x) = 2 \frac{d}{dx} K(x, x)$. Israel Gelfand and Boris Levitan introduced it, and the related Marchenko equation is a basic tool of the inverse scattering transform.