---
tag: MW5R19P2X4
kind: example
name: Lippmann–Schwinger equation
msc: [81U05, 35P25, 45A05]
instance_of: [8KJCN49V64]
sources:
  - url: https://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation
same_as:
  wikidata: Q902575
  wikipedia: Lippmann–Schwinger equation
---

## Example

Consider the stationary [[Schrödinger equation|EPZR6H2WJN]] $-\Delta \psi + V \psi = k^2 \psi$ on $\R^3$, with a real potential $V$ of compact support, in units with $\hbar^2 / 2m = 1$.
Let $k > 0$ and let $\mathbf{k} \in \R^3$ with $\abs{\mathbf{k}} = k$ be the direction and wavenumber of an incoming plane wave.
The scattering solution $\psi$ solves the Lippmann–Schwinger equation
$$\psi(x) = e^{i \mathbf{k} \cdot x} - \frac{1}{4\pi} \int_{\R^3} \frac{e^{ik\abs{x - y}}}{\abs{x - y}} \, V(y) \, \psi(y) \, dy, \quad x \in \R^3.$$
It is an [[integral equation of the second kind|8KJCN49V64]] for $\psi$, with the [[weakly singular kernel|94TNBAAJYX]] $-\frac{1}{4\pi} \frac{e^{ik\abs{x - y}}}{\abs{x - y}} V(y)$.

Check: the function $G(x) = -\frac{e^{ik\abs{x}}}{4\pi \abs{x}}$ is the outgoing [[Green's function|SH6JY9RN2E]] of the Helmholtz operator, $(\Delta + k^2) G = \delta_0$.
Since $(\Delta + k^2) e^{i\mathbf{k} \cdot x} = 0$, applying $\Delta + k^2$ to the right side gives $V \psi$, which is the Schrödinger equation $(\Delta + k^2)\psi = V \psi$.
The choice of $e^{+ik\abs{x - y}}$ makes the scattered part $\psi - e^{i \mathbf{k} \cdot x}$ an outgoing spherical wave at infinity.

## Remarks

In operator form the equation reads $\psi = \phi + G_0 V \psi$ with $\phi$ the incoming wave and $G_0$ the convolution with $G$.
Iterating it gives the [[Born series|2PCVRG9NBN]], and its first term beyond $\phi$ is the Born approximation.

## Explanations

### L1 human

A wave hits an obstacle described by $V$ and scatters.
The equation says that the total wave is the incoming wave plus the waves sent out by each point of the obstacle.
Each point sends out a wave with strength equal to the potential times the total wave there.
Since the unknown wave appears on both sides, it is an integral equation rather than a formula.