Ebook: Inverse Problems in Quantum Scattering Theory

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.

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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.

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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.
Content:
Front Matter....Pages i-xxxi
Some Results from Scattering Theory....Pages 1-19
Bound States—Eigenfunction Expansions....Pages 20-39
The Gel’fand-Levitan-Jost-Kohn Method....Pages 40-58
Applications of the Gel’fand-Levitan Equation....Pages 59-75
The Marchenko Method....Pages 76-85
Examples....Pages 86-94
Special Classes of Potentials....Pages 95-111
Nonlocal Separable Interactions....Pages 112-127
Miscellaneous Approaches to the Inverse Problems at Fixed l ....Pages 128-154
Scattering Amplitudes from Elastic Cross Sections....Pages 155-181
Potentials from the Scattering Amplitude at Fixed Energy: General Equation and Mathematical Tools....Pages 182-194
Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods....Pages 195-213
Potentials from the Scattering Amplitude at Fixed Energy: Operator Methods....Pages 214-234
The Three-Dimensional Inverse Problem....Pages 235-274
Miscellaneous Approaches to Inverse Problems at Fixed Energy....Pages 275-289
Approximate Methods....Pages 290-322
Inverse Problems in One Dimension....Pages 323-388
Problems Connected with Discrete Spectra....Pages 389-415
Numerical Problem....Pages 416-439
Back Matter....Pages 441-499

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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.
Content:
Front Matter....Pages i-xxxi
Some Results from Scattering Theory....Pages 1-19
Bound States—Eigenfunction Expansions....Pages 20-39
The Gel’fand-Levitan-Jost-Kohn Method....Pages 40-58
Applications of the Gel’fand-Levitan Equation....Pages 59-75
The Marchenko Method....Pages 76-85
Examples....Pages 86-94
Special Classes of Potentials....Pages 95-111
Nonlocal Separable Interactions....Pages 112-127
Miscellaneous Approaches to the Inverse Problems at Fixed l ....Pages 128-154
Scattering Amplitudes from Elastic Cross Sections....Pages 155-181
Potentials from the Scattering Amplitude at Fixed Energy: General Equation and Mathematical Tools....Pages 182-194
Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods....Pages 195-213
Potentials from the Scattering Amplitude at Fixed Energy: Operator Methods....Pages 214-234
The Three-Dimensional Inverse Problem....Pages 235-274
Miscellaneous Approaches to Inverse Problems at Fixed Energy....Pages 275-289
Approximate Methods....Pages 290-322
Inverse Problems in One Dimension....Pages 323-388
Problems Connected with Discrete Spectra....Pages 389-415
Numerical Problem....Pages 416-439
Back Matter....Pages 441-499
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