Potentials from the Scattering Amplitude at Fixed Energy: Operator Methods

Abstract

In matrix methods, the goal was to construct potentials in a large parametrized class of functions, then to identify this class. A different point of view is to define a certain large class of functions a priori and then try to answer the questions of section XI.1 (existence, uniqueness, construction ...) in this class. In the first kind of approach, one risks obtaining potentials that violate certain fine physical requirements. In the second kind, one risks obtaining poor or very partial answers to the question C and the following questions (constructibility, stability ...). This remark is verified in the two first approaches given in this chapter. In the third approach, the combination of the two points of view enables one to answer all questions, but the scattering equivalent classes are large, and identifying special physical classes is still a problem.