Scattering compares asymptotic free states at $t\to-\infty$ and $t\to+\infty$. The $S$-matrix maps in-states to out-states, and the $T$-matrix encodes interaction-induced transitions.

For central potentials, the scattering wave is expanded in partial waves labeled by $\ell$. The phase shift $\delta_\ell$ determines the contribution of each angular momentum channel and reveals resonant behavior.

Unitarity of the $S$-matrix implies probability conservation and yields the optical theorem, which relates the total cross section to the forward scattering amplitude. This is a powerful consistency check in both nonrelativistic and relativistic scattering.

At weak coupling, the first Born approximation gives the scattering amplitude as the Fourier transform of the potential, up to conventional factors. It is often a starting point for more refined multiple-scattering treatments.

The exact scattering state satisfies $|\psi^{(+)}\rangle=|\phi\rangle+G_0^{(+)}V|\psi^{(+)}\rangle$. This integral equation underlies Green's-function methods, diagrammatic expansions, and numerical solvers.