Wave Scattering by Time-Dependent Perturbations: An Introduction

Preface ix

Chapter 1: Introduction and Outline of Contents 1
1.1 Introduction 1
1.2 Some Illustrations 4
1.3 Towards Generalisations 6
1.4 Chapter Summaries 23

Chapter 2: Some Aspects of Waves on Strings 26
2.1 Introduction 26
2.2 A Free Problem 27
2.3 On Solutions of the Wave Equation 28
2.4 On Solutions of Initial Value Problems for the Wave Equation 31
2.5 Integral Transform Methods 33
2.6 Reduction to a First-Order System 37
2.7 Some Perturbed Problems for Waves on Strings 39
2.7.1 Waves on a Nonuniform String 39
2.7.2 Waves on a Semi-infinite String with a Fixed End 42
2.7.3 Waves on an Elastically Braced String 45
2.7.4 Waves on a String of Varying Length 47
2.7.5 A Scattering Problem on a Semi-infinite String 49

Chapter 3: Mathematical Preliminaries 55
3.1 Introduction 55
3.2 Notation 55
3.3 Vector Spaces 56
3.4 Distributions 62
3.5 Fourier Transforms and Distributions 72
3.6 Hilbert Spaces 79
3.6.1 Orthogonality, Bases and Expansions 85
3.6.2 Linear Functionals and Operators on Hilbert Spaces 92
3.6.3 Some Frequently Occurring Operators 98
3.6.4 Unbounded Linear Operators on Hilbert Spaces 104
3.6.5 Some Remarks Concerning Unbounded Operators 108

Chapter 4: Spectral Theory and Spectral Decompositions 113
4.1 Introduction 113
4.2 Basic Concepts 113
4.3 Concerning Spectral Decompositions 118
4.3.1 Spectral Decompositions on Finite-Dimensional Spaces 119
4.3.2 Reducing Subspaces 124
4.3.3 Spectral Decompositions on Infinite-Dimensional Spaces 128
4.4 Some Properties of Spectral Families 131
4.5 Concerning the Determination of Spectral Families 132
4.6 On Functions of an Operator 135
4.7 Spectral Decompositions of Hilbert Spaces 138
4.7.1 An Illustration 138
4.7.2 A Little More About Riemann-Stieltjes Integrals 139
4.7.3 Spectral Measure 141
4.7.4 On Spectral Subspaces of a Hilbert Space 143

Chapter 5: On Nonautonomous Problems 146
5.1 Introduction 146
5.2 Concerning Semigroup Methods 146
5.2.1 On the Well-posedness of Problems 150
5.2.2 On Generators of Semigroups 151
5.3 The Propagator and Its Properties 155
5.4 On the Solution of a Nonautonomous Wave Problem 159
5.4.1 A Mathematical Model 159
5.4.2 Energy Space Setting and Solution Concepts 160
5.4.3 Reduction to a First-Order System 161
5.4.4 On the Construction of the Propagator and the Solution 162
5.5 Some Results from the Theory of Integral Equations 163

Chapter 6: On Scattering Theory Strategies 174
6.1 Introduction 174
6.2 On Scattering Processes in Autonomous Problems 174
6.2.1 Propagation Aspects 175
6.2.2 Solutions with Finite Energy and Scattering States 180
6.2.3 On the Construction of Solutions 182
6.2.4 Wave Operators and Their Construction 185
6.2.5 More About Asymptotic Conditions 191
6.2.6 A Remark About Spectral Families 196
6.2.7 Some Comparisons of the Two Approaches 196
6.2.8 Summary 199
6.3 On Scattering Processes in Nonautonomous Problems 199
6.3.1 Propagation Aspects 200
6.3.2 Scattering Aspects 201
6.3.3 On the Construction of Propagators and Solutions 202

Chapter 7: Echo Analysis 209
7.1 Introduction 209
7.2 Concerning the Mathematical Model 209
7.3 Scattering Aspects and Echo Analysis 213
7.4 On the Construction of the Echo Field 214
7.4.1 Zero Approximation for the Echo Field 218
7.4.2 Concerning Higher-Order Approximations 223
7.5 A Remark About Energy in the System 224

Chapter 8: Wave Scattering from Time-Periodic Perturbations 225
8.1 Introduction 225
8.2 Concerning the Mathematical Model 225
8.3 Basic Assumptions, Definitions and Results 226
8.4 Some Remarks on Estimates for Propagators 231
8.5 Scattering Aspects 231
8.5.1 Some Results for Potential Scattering 233

Chapter 9: Concerning Inverse Problems 235
9.1 Introduction 235
9.2 Preliminaries 236
9.3 Reduction of the Plasma Wave Equation to a First-Order System 239
9.4 A High-Energy Method 240
9.4.1 Some Asymptotic Formulae for the Plasma Wave Equation 240
9.4.2 On the Autonomous Inverse Scattering Problem 243
9.4.3 Extension to Nonautonomous Inverse Scattering Problems 244

Chapter 10: Some Remarks on Scattering in Other Wave Systems 246
10.1 Introduction 246
10.2 Scattering of Electromagnetic Waves 246
10.3 Strategy for Autonomous Acoustics Problems in R3 253
10.4 Strategies for Electromagnetic Scattering Problems 256
10.4.1 Concerning Autonomous Problems 256
10.4.2 Concerning Nonautonomous Problems 259
10.5 Scattering of Elastic Waves 259
10.5.1 Strategy for Autonomous Elastic Wave Scattering Problems 259

Chapter 11: Commentaries and Appendices 263
11.1 Remarks on Previous Chapters 263
11.2 Appendices 266

Bibliography 275
Index 285