Editing Scattering (section)

===Mathematical framework===
In [[mathematics]], scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a [[differential equation]] is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of [[time]].  One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made  to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future".  The scattering matrix then pairs solutions in the "distant past" to those in the "distant future".

Solutions to differential equations are often posed on [[manifold]]s. Frequently, the means to the solution requires the study of the [[Spectrum (functional analysis)|spectrum]] of an [[operator theory|operator]] on the manifold. As a result, the solutions often have a spectrum that can be identified with a [[Hilbert space]], and scattering is described by a certain map, the [[S matrix]], on Hilbert spaces.  Solutions with a [[discrete spectrum (physics)|discrete spectrum]] correspond to [[bound state]]s in quantum mechanics, while a [[continuous spectrum]] is associated with scattering states.  The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.

An important, notable development is the [[inverse scattering transform]], central to the solution of many [[exactly solvable model]]s.