Born series

Summary

The Born series is the expansion of a scattered wave in powers of the scattering potential. It is obtained by iterating the Lippmann–Schwinger equation $\psi = \phi + G_0 V \psi$, which gives $\psi = \phi + G_0 V \phi + G_0 V G_0 V \phi + \cdots$. It is the Liouville–Neumann series of this integral equation. So it converges when the operator $G_0 V$ is small, for example for weak potentials or at high energy. Its first term beyond $\phi$ is the Born approximation, the standard first estimate of scattering amplitudes.