Inverse Problems in Scattering: An Introduction [Paperback]

Inverse Problems in Scattering exposes some of the mathematics which has been developed in attempts to solve the one-dimensional inverse scattering problem. Layered media are treated in Chapters 1--6 and quantum mechanical models in Chapters 7--10. Thus, Chapters 2 and 6 show the connections between matrix theory, Schur's lemma in complex analysis, the Levinson--Durbin algorithm, filter theory, moment problems and orthogonal polynomials. The chapters devoted to the simplest inverse scattering problems in quantum mechanics show how the Gel'fand--Levitan and Marchenko equations arose. The introduction to this problem is an excursion through the inverse problem related to a finite difference version of Schr?dinger's equation. One of the basic problems in inverse quantum scattering is to determine what conditions must be imposed on the scattering data to ensure that they correspond to a regular potential, which involves Lebesque integrable functions, which are introduced in Chapter 9.
1 Some Simple Wave Phenomena.- 1.1 Waves.- 1.2 Generalized Functions.- 1.3 Lossless Transmission Lines.- 1.4 Waves in an Elastic Medium.- 1.5 Sound Waves.- 1.6 Reflection and Transmission of Waves.- 1.7 Standardising the Wave Equations.- 1.8 Standardised Reflection and Transmission.- 1.9 Conservation Equations.- 1.10 Difference Schemes for Wave Problems.- 1.11 Down-Up Difference Schemes.- 1.12 Causal Solutions, Greens Functions and z-Transforms.- 2 Layer-Peeling Methods for Discrete Inverse Problems.- 2.1 Introduction.- 2.2 On the Interpretation of the Discrete Equations.- 2.3 The Impulse Response and the Propagation of the Wavefront.- 2.4 Discrete Equations for Layer-Peeling Methods.- 2.5 Some Questions.- 2.6 Functions of a Complex Variable.- 2.7 Schurs Lemma.- 2.8 The Schur Algorithm and Downward Continuation.- 2.9 The Schur Algorithm and Toeplitz Matrices.- 2.10 Layered Media of Non-Goupillaud Type.- 3 The Inversion of Discrete Systems Usilˆ