Scattering Theory

Scattering theory is a subset of spectral theory that seeks to parametrize the absolutely continuous spectrum of an elliptic operator H on a complete manifold M with uniform structure at infinity. In the simplest case where M is compact, the spectrum of H is discrete, and scattering theory is trivial. On R, scattering theory is already non-trivial, and the techniques used in studying scattering on R can be generalized to non-Euclidean manifolds. In particular, perturbation theory is used to observe the behavior of a full Hamiltonian H relative to a free Hamiltonian H0. The main two problems that scattering theory is concerned with are investigating the asymptotics as t→ ±∞ of the time-dependent Schrodinger equation