The 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 manifolds. Frequently, the means to the solution requires the study of the spectrum of an 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. 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.
Read more about this topic: Scattering Theory
Famous quotes containing the words mathematical and/or framework:
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“Most young black females learn to be suspicious and critical of feminist thinking long before they have any clear understanding of its theory and politics.... Without rigorously engaging feminist thought, they insist that racial separatism works best. This attitude is dangerous. It not only erases the reality of common female experience as a basis for academic study; it also constructs a framework in which differences cannot be examined comparatively.”
—bell hooks (b. c. 1955)