Inverse Problems in Quantum Scattering Theory

The physical importance of inverse problems in quantum scattering theory is clear since all the information we can obtain on nuclear, particle, and subparticle physics is gathered from scattering experiments. Exact and approximate methods of investigating scattering theory, inverse radial problems at fixed energy, inverse one-dimensional problems, inverse three-dimensional problems, and construction of the scattering amplitude from the cross section are presented. The methods often apply to other fields, e.g. applied mathematics and geophysics. The book will therefore be of interest to theoretical and mathematical physicists, nuclear particle physicists, and chemical physicists. For the second edition the chapters on one-dimensional and three-dimensional scattering problems have been rewritten and considerably expanded. Furthermore, two new chapters on spectral problems and on numerical aspects have been added; in the sections on classical methods the comments and references have been updated.

The normal business of physicists may be schematically thought of as predic- ting the motions of particles on the basis of known forces, or the propagation of radiation on the basis of a known constitution of matter. The inverse problem is to conclude what the forces or constitutions are on the basis of the observed motion. A large part of our sensory contact with the world around us depends on an intuitive solution of such an inverse problem: We infer the shape, size, and surface texture of external objects from their scattering and absorption of light as detected by our eyes. When we use scattering experiments to learn the size or shape of particles, or the forces they exert upon each other, the nature of the problem is similar, if more refined. The kinematics, the equations of motion, are usually assumed to be known. It is the forces that are sought, and how they vary from point to point. As with so many other physical ideas, the first one we know of to have touched upon the kind of inverse problem discussed in this book was Lord Rayleigh (1877). In the course of describing the vibrations of strings of variable density he briefly discusses the possibility of inferring the density distribution from the frequencies of vibration. This passage may be regarded as a precursor of the mathematical study of the inverse spectral problem some seventy years later.

Contents: Some Results from Scattering Theory.- Bound States - Eigenfunction Expansions.- The Gel'fand-Levitan-Jost-Kohn Method.- Applications of the Gel'fand-Levitan Equation.- The Marchenko Method.- Examples.- Special Classes of Potentials.- Nonlocal Separable Interactions.- Miscellaneous Approaches to the Inverse Problems at Fixed l.- Scattering Amplitudes from Elastic Cross Sections.- Potentials from the Scattering Amplitude at Fixed Energy: General Equation and Mathematical Tools.- Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods.- Potentials from the Scattering Amplitude at Fixed Energy: Operator Methods.- The Three-Dimensional Inverse Problem.- Miscellaneous Approaches to Inverse Problems at Fixed Energy.- Approximate Methods.- Inverse Problems in One Dimension.- Problems Connected with Discrete Spectra.- Numerical Problem.- Reference List.- Subject Index.