# inverse scattering theory

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## Inverse scattering theory

A theory whose objective is to determine the scattering object, or an interaction potential energy, from the knowledge of a scattered field. This is the opposite problem from direct scattering theory, where the scattering amplitude is determined from the equations of motion, including the potential. The equations of motion are usually linear (operator-valued) equations. *See* Scattering experiments (atoms and molecules), Scattering experiments (nuclei)

Inverse scattering theories can be divided into two types: (1) pure inverse problems, when the data consist of complete, noise-free information of the scattering amplitude; and (2) applied inverse problems, when incomplete data which are corrupted by noise are given. Many different applied inverse problems can be obtained from any pure inverse problem by using different band-limiting procedures and different noise spectra.

The difficulty of determining the exact object which produced a scattering amplitude is evident. It is often a priori information about the scatterer that makes the inversion possible.

Much of the basic knowledge of systems of atoms, molecules, and nuclear particles is obtained from inverse scattering studies using beams of different particles as probes. For the Schrödinger equation with spherical symmetry or in one dimension, there is an exact solution of the inverse problem.

A number of high-technology areas (nondestructive evaluation, medical diagnostics including acoustic and ultrasonic imaging, x-ray absorption and nuclear magnetic resonance tomography, radar scattering and geophysical exploration) use inverse scattering theory. Several classical waves including acoustic, electromagnetic, ultrasonic, x-rays, and others are used.

All of the inverse scattering technologies require the solution to ill-posed or improperly posed problems. A model equation is well posed if it has a unique solution which depends continuously on the initial data. It is ill posed otherwise. The ill-posed problems which are amenable to analysis, called regularizable ill-posed problems, are those which depend discontinuously upon the data. This destroys uniqueness, although solutions (in fact, many solutions) exist.