Scattering Theory, Revised Edition (eBook)

Front Cover 1
Scattering Theory 6
Copyright Page 7
Contents 8
Preface to the Revised Edition 10
Preface to the First Edition 11
Chapter I. Introduction 14
1. The Dynamic Approach 17
2. Scattering Theory Formulated in Terms of Representation Theory 19
3. A Semigroup of Operators Related to the Scattering Matrix 24
4. The Form of the Scattering Matrix 28
5. A Simple Example 31
6. Scattering Theory for Transport Phenomena 40
7. Notes and Remarks 48
Chapter II. Representation Theory and the Scattering Operator 50
1. The Discrete Case 50
2. The Scattering Operator in the Discrete Case 54
3. The Continuous Case 59
4. The Scattering Operator in the Continuous Case 64
5. Notes and Remarks 71
Chapter III. A Semigroup of Operators Related to the Scattering Matrix 74
1. The Related Semigroups 75
2. On Semigroups of Contraction Operators 79
3. Spectral Theory 81
4. A Spectral Mapping Theorem 90
5. Applications of the Spectral Theory 96
6. Equivalent Incoming and Outgoing Representations 100
7. Notes and Remarks 104
Chapter IV. The Translation Representation for the Solution of the Wave Equation in Free Space 106
1. The Hilbert Space Ho and the Group {U0(t)} 107
2. Spectral and Translation Representations of {U0(t)} 112
3. The Operator I Extended to Distributions 122
4. Translation Representation for Outgoing and Incoming Data with Infinite Energy 131
5. Notes and Remarks 143
Chapter V. The Solution of the Wave Equation in an Exterior Domain 146
1. The Hilbert Space H and the Group {U(t)} 148
2. Energy Decay and Translation Representations 155
3. The Semigroup {Z(t)} 164
4. The Relation between the Semigroup {Z(t) } and the solutions of the Reduced Wave Equation 171
5. The Scattering Matrix 178
6. Notes and Remarks 188
Chapter VI. Symmetric Hyperbolic Systems, the Acoustic Equation with an Indefinite Energy Form, and the Schrödinger Equation 190
Part 1. Symmetric Hyperbolic Systems 191
1. Translation Representation in Free Space 191
2. Solutions of Hyperbolic Systems in an Exterior Domain 210
Part 2. The Acoustic Equation with an Indefinite Energy Form and the Schrödmger Equation 232
3. Scattering for the Acoustic Equation with an Indefinite Energy Form 232
4. The Schrödinger Scattering Matrix 253
5. Notes and Remarks 257
Appendix 1. Semigroups of Operators 258
Appendix 2. Energy Decay 268
Appendix 3. Energy Decay for Star-Shaped Obstacles 274
Appendix 4. Scattering Theory for Maxwell's Equations 278
References 282
Epilogue 288
Index 320
Pure and Applied Mathematics 324