An Introduction to Linear and Nonlinear Scattering Theory  book cover
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An Introduction to Linear and Nonlinear Scattering Theory





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ISBN 9780367448264
September 30, 2020 Forthcoming by Routledge
264 Pages

 
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Book Description

This monograph has two main purposes, first to act as a companion volume to more advanced texts by gathering together the principal mathematical topics commonly used in developing scattering theories and, in so doing, provide a reasonable, self-contained introduction to linear and nonlinear scattering theory for those who might wish to begin working in the area. Secondly, to indicate how these various aspects might be applied to problems in mathematical physics and the applied sciences. Of particular interest will be the influence of boundary conditions.

Table of Contents

Preface -- Chapter 1: Introduction and outline of contents -- Chapter 2: Analytical preliminaries -- 2.1 Introduction -- 2.2 Preliminaries -- 2.3 Distribution theory -- 2.4 Hilbert space -- 2.5 Bounded linear operators on Hilbert spaces -- 2.6 Unbounded linear operators on Hilbert spaces -- 2.7 Adjoints of unbounded operators -- 2.8 A basic criterion for self-adjointness -- Chapter 3: Examples of scattering theory strategies -- 3.1 Introduction -- 3.2 A free problem -- 3.3 Properties of the operators Ao and Uo(t) -- 3.4 A perturbed problem -- 3.5 Comparison of solutions -- 3.6 Summary -- Chapter 4: Elements of spectral theory -- 4.1 Introduction -- 4.2 Basic concepts -- 4.3 Eigenvalues and eigenvectors -- 4.4 Spectral decompositions on finite dimensional spaces -- 4.5 Spectral decomposition on infinite dimensional spaces -- 4.6 Properties of spectral families -- 4.7 Functions of an operator -- 4.8 Spectral decompositions of H -- 4.9 Examples -- 4.10 More on spectral decompositions associated with an operator -- 4.11 On the determination of spectral families -- Chapter 5: Some applications of semigroup theory -- 5.1 Introduction and basic results -- 5.2 On the well-posedness of problems -- 5.3 Generators of semigroups -- 5.4 Perturbation of semigroups -- Chapter 6: More about wave operators -- 6.1 Introduction -- 6.2 Abstract evolutionary systems -- 6.3 The scattering operator -- 6.4 Existence of wave operators -- 6.5 Selected properties of wave operators -- 6.6 On the completeness of wave operators -- Chapter 7: Target scattering -- 7.1 Introduction -- 7.2 Concerning the incident field -- 7.3 A typical target scattering problem -- 7.4 Solution concepts -- 7.5 Concerning existence and uniqueness of solutions -- Chapter 8: A scattering theory -- 8.1 Introduction -- 8.2 A free problem -- 8.3 A perturbed problem -- 8.4 Concerning the wave operators -- 8.5 Summary and additional comments -- Chapter 9: Nonlinear scattering theory -- 9.1 Introduction -- 9.2 Concerning existence of solutions -- 9.3 Scattering theory -- 9.4 More on conditions ensuring scattering -- Chapter 10: Commentaries -- References -- Index.

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Author(s)

Biography

Roach, G F

Reviews

"Roach has taken on a difficult task and done a good job. We should all be thankful to him for having done so."
-SIAM Rev. Vol 39 No 2