Wave Scattering by Time-Dependent Perturbations: An Introduction (Princeton Series in Applied Mathematics)

0691113408......Page 1
Contents......Page 6
Preface......Page 10
1.1 Introduction......Page 14
1.2 Some Illustrations......Page 17
1.3 Towards Generalisations......Page 19
1.4 Chapter Summaries......Page 36
2.1 Introduction......Page 39
2.2 A Free Problem......Page 40
2.3 On Solutions of the Wave Equation......Page 41
2.4 On Solutions of Initial Value Problems for the Wave Equation......Page 44
2.5 Integral Transform Methods......Page 46
2.6 Reduction to a First-Order System......Page 50
2.7.1 Waves on a Nonuniform String......Page 52
2.7.2 Waves on a Semi-infinite String with a Fixed End......Page 55
2.7.3 Waves on an Elastically Braced String......Page 58
2.7.4 Waves on a String of Varying Length......Page 60
2.7.5 A Scattering Problem on a Semi-infinite String......Page 62
3.2 Notation......Page 68
3.3 Vector Spaces......Page 69
3.4 Distributions......Page 75
3.5 Fourier Transforms and Distributions......Page 85
3.6 Hilbert Spaces......Page 92
3.6.1 Orthogonality, Bases and Expansions......Page 98
3.6.2 Linear Functionals and Operators on Hilbert Spaces......Page 105
3.6.3 Some Frequently Occurring Operators......Page 111
3.6.4 Unbounded Linear Operators on Hilbert Spaces......Page 117
3.6.5 Some Remarks Concerning Unbounded Operators......Page 121
4.2 Basic Concepts......Page 126
4.3 Concerning Spectral Decompositions......Page 131
4.3.1 Spectral Decompositions on Finite-Dimensional Spaces......Page 132
4.3.2 Reducing Subspaces......Page 137
4.3.3 Spectral Decompositions on Infinite-Dimensional Spaces......Page 141
4.4 Some Properties of Spectral Families......Page 144
4.5 Concerning the Determination of Spectral Families......Page 145
4.6 On Functions of an Operator......Page 148
4.7.1 An Illustration......Page 151
4.7.2 A Little More About Riemann-Stieltjes Integrals......Page 152
4.7.3 Spectral Measure......Page 154
4.7.4 On Spectral Subspaces of a Hilbert Space......Page 156
5.2 Concerning Semigroup Methods......Page 159
5.2.1 On the Well-posedness of Problems......Page 163
5.2.2 On Generators of Semigroups......Page 164
5.3 The Propagator and Its Properties......Page 168
5.4.1 A Mathematical Model......Page 172
5.4.2 Energy Space Setting and Solution Concepts......Page 173
5.4.3 Reduction to a First-Order System......Page 174
5.4.4 On the Construction of the Propagator and the Solution......Page 175
5.5 Some Results from the Theory of Integral Equations......Page 176
6.2 On Scattering Processes in Autonomous Problems......Page 187
6.2.1 Propagation Aspects......Page 188
6.2.2 Solutions with Finite Energy and Scattering States......Page 193
6.2.3 On the Construction of Solutions......Page 195
6.2.4 Wave Operators and Their Construction......Page 198
6.2.5 More About Asymptotic Conditions......Page 204
6.2.7 Some Comparisons of the Two Approaches......Page 209
6.2.8 Summary......Page 212
6.3.1 Propagation Aspects......Page 213
6.3.2 Scattering Aspects......Page 214
6.3.3 On the Construction of Propagators and Solutions......Page 215
7.2 Concerning the Mathematical Model......Page 222
7.3 Scattering Aspects and Echo Analysis......Page 226
7.4 On the Construction of the Echo Field......Page 227
7.4.1 Zero Approximation for the Echo Field......Page 231
7.4.2 Concerning Higher-Order Approximations......Page 236
7.5 A Remark About Energy in the System......Page 237
8.2 Concerning the Mathematical Model......Page 238
8.3 Basic Assumptions, Definitions and Results......Page 239
8.5 Scattering Aspects......Page 244
8.5.1 Some Results for Potential Scattering......Page 246
9.1 Introduction......Page 248
9.2 Preliminaries......Page 249
9.3 Reduction of the Plasma Wave Equation to a First-Order System......Page 252
9.4.1 Some Asymptotic Formulae for the Plasma Wave Equation......Page 253
9.4.2 On the Autonomous Inverse Scattering Problem......Page 256
9.4.3 Extension to Nonautonomous Inverse Scattering Problems......Page 257
10.2 Scattering of Electromagnetic Waves......Page 259
10.3 Strategy for Autonomous Acoustics Problems in R3......Page 266
10.4.1 Concerning Autonomous Problems......Page 269
10.5.1 Strategy for Autonomous Elastic Wave Scattering Problems......Page 272
11.1 Remarks on Previous Chapters......Page 276
11.2 Appendices......Page 279
Bibliography......Page 288
Index......Page 298