---
tag: X35RWR285S
kind: concept
name: Gelfand–Levitan–Marchenko equation
msc: [34A55, 34L25, 45B05]
sources:
  - url: https://en.wikipedia.org/wiki/Marchenko_equation
same_as:
  wikidata: Q17378628
  wikipedia: Marchenko equation
---

## Summary

The Gelfand–Levitan–Marchenko equation is a linear [[integral equation|R3SG1SN7JQ]] 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|6GY3X4Y8X5]], which solves the [[Korteweg–De Vries equation|KTT75QC72Z]] and other integrable equations.

## Explanations

### L1 human

Send waves at an unknown bump in a potential and record what comes back: how much of each wave is reflected and which waves stay trapped.
The inverse problem asks for the bump from these echoes.
The Gelfand–Levitan–Marchenko equation turns this question into a linear integral equation, which is much easier than the nonlinear problem it replaces.
When there is no reflection and one trapped state, the equation can be solved by hand and gives the potential $-2\kappa^2 \operatorname{sech}^2(\kappa(x - x_0))$, the shape of a [[soliton|63D4AXYCT4]].