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When two atoms are scattered off one another, one can understand them as being the bound state solutions of some differential equation. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even ionized , representing an inelastic scattering process.

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The term " deep inelastic scattering " refers to a special kind of scattering experiment in particle physics. 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 manifolds. Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold.

## Scattering Theory - 1st Edition

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. Spaces with a discrete spectrum correspond to bound states 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 models.

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Part of a series on. Introduction Glossary History. Access provided by: anon Sign Out. Recursive limited memory filtering and scattering theory Corresp. Abstract: Redheffer scattering theory is reviewed in a generalized setting as a method to derive recursive solutions of linear two-point boundary value problems TPBVP over arbitrarily varying intervals.

The results can be used to derive a complete solution for the problem of limited-memory or sliding-window estimation, when a usual state-space model for the signal is available. Recursive limited-memory filters are derived for both continuous and discrete time signals.

Article :. Date of Publication: May