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An Introduction to Inverse Scattering and Inverse Spectral Problems

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by Khosrow Chadan, David Colton, Lassi Päivärinta, and William Rundell


1997 / x + 198 pages / Softcover / ISBN: 978-0-898713-87-9 / List Price $80.00 / SIAM Member Price $56.00 / Order Code MM02

Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.

In the opening chapter, PŠivŠrinta collects the mathematical tools needed in the subsequent chapters and gives references for further study. Colton's chapter focuses on electromagnetic scattering problems. As an application he considers the problem of detecting and monitoring leukemia. Rundell's chapter deals with inverse spectral problems. He describes several exact and algorithmic methods for reconstructing an unknown function from the spectral data. Chadan provides an introduction to quantum mechanical inverse scattering problems. As an application he explains the celebrated method of Gardner, Greene, Kruskal, and Miura for solving nonlinear evolution equations such as the Korteweg_de Vries equation. Each chapter provides full references for further study.

The book is written for graduate students, applied mathematicians, and electrical engineers interested in scattering theory and inverse problems. It could be used in a special inverse problems course or as background material for PDE or functional analysis courses. A background in elementary functional analysis and analytic function theory is needed.

Foreword; Preface; Chapter 1: A Review of Basic Mathematical Tools, Lassi PŠivŠrinta. Linear Operators on Hilbert Space; Integral Operators and the Fredholm Alternative; The Fourier Transform and the Hilbert Transform; The Unique Continuation Principle (UCP); Unbounded Operators; The Spectrum; The Resolvent Kernel and the Fredholm Determinant; A Particle in a Box; Maxwell's Equations; References; Chapter 2: Multidimensional Inverse Scattering Theory, David Colton. Electromagnetic Scattering Problem; Bessel Functions; The Addition Formula; Green's Formula; Basic Properties of Far Field Patterns; Spectral Theory of the Far Field Operator; The Inverse Scattering Problem; The Detection and Monitoring of Leukemia; Regularization; Closing Remarks; References; Chapter 3: Inverse SturmÐLiouville Problems, William Rundell. Introduction; Preliminary Material; The Liouville Transformation; Asymptotic Expansions of the Eigenvalues and Eigenfunctions; The Inverse ProblemÑA Historical Look; A Completeness Result; An Important Integral Operator; Solving Hyperbolic Equations; Uniqueness Proofs; Constructive Algorithms; Modification for Other Spectral Data; Other Differential Equations; Other Constructive Algorithms; The Matrix Analogue; Another Finite-Dimensional Algorithm; Fourth-Order Problems; References; Chapter 4: Inverse Problems in Potential Scattering, Khosrow Chadan. Introduction; Physical Background and Formulation of the Inverse Scattering Problem; Scattering Theory for Partial Waves; Gel'fandÐLevitan Integral Equation; Marchenko Equation; Inverse Problem on the Line; Nonlinear Evolution Equations; Closing Remarks; Appendix; References; Index.

ISBN: 9780898713879

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