An Introduction to Inverse Scattering and Inverse Spectral Methods
Published on 31. July 1997
Book
Paperback/Softback
208 pages
978-0-89871-387-9 (ISBN)
Description
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, Paeivaerinta 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.
In the opening chapter, Paeivaerinta 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.
More details
Series
Language
English
Place of publication
New York
United States
Target group
College/higher education
Product notice
Paperback (trade)
Dimensions
Height: 255 mm
Width: 175 mm
Thickness: 12 mm
Weight
372 gr
ISBN-13
978-0-89871-387-9 (9780898713879)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
Foreword
Preface
Chapter 1: A Review of Basic Mathematical Tools, Lassi Paeivaerinta. 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 Sturm-Liouville Problems, William Rundell. Introduction
Preliminary Material
The Liouville Transformation
Asymptotic Expansions of the Eigenvalues and Eigenfunctions
The Inverse Problem-A 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'fand-Levitan Integral Equation
Marchenko Equation
Inverse Problem on the Line
Nonlinear Evolution Equations
Closing Remarks
Appendix
References
Index.
Preface
Chapter 1: A Review of Basic Mathematical Tools, Lassi Paeivaerinta. 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 Sturm-Liouville Problems, William Rundell. Introduction
Preliminary Material
The Liouville Transformation
Asymptotic Expansions of the Eigenvalues and Eigenfunctions
The Inverse Problem-A 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'fand-Levitan Integral Equation
Marchenko Equation
Inverse Problem on the Line
Nonlinear Evolution Equations
Closing Remarks
Appendix
References
Index.