1. Conventions .................................................. 1
2. Mathematical Introduction .................................... 5
2.1. Quantum Theory .......................................... 5
2.2. Wave Equations .......................................... 8
3. Prerequisites ............................................... 13
3.1. Linear Operators in Banach Spaces ...................... 13
3.2. Weak Integration of Banach Space-Valued Maps ........... 25
3.3. Exponentials of Bounded Linear Operators ............... 35
4. Strongly Continuous Semigroups .............................. 41
4.1. Elementary Properties .................................. 42
4.2. Characterizations ...................................... 51
4.3. An Integral Representation in the Complex Case ......... 58
4.4. Perturbation Theorems .................................. 59
4.5. Strongly Continuous Groups ............................. 63
4.6. Associated Inhomogeneous Initial Value Problems ........ 66
5. Examples of Generators of Strongly Continuous Semigroups .... 71
5.1. The Ordinary Derivative on a Bounded Interval .......... 71
5.2. Linear Stability of Ideal Rotating Couette Flows ....... 74
5.3. Outgoing Boundary Conditions ........................... 77
5.4. Damped Wave Equations .................................. 84
5.5. Autonomous Linear Hemiitian Hyperbolic Systems ......... 97
6. Intertwining Relations, Operator Homomorphisms ............. 105
6.1. Semigroups and Their Restrictions ..................... 105
6.2. Intertwining Relations ................................ 114
6.3. Nonexpansive Homomorphisms ............................ 117
7. Examples of Constrained Systems ............................ 123
7.1. 1-D Wave Equations with Sommerfeld Boundary
Conditions ............................................ 123
7.2. 1-D Wave Equations with Engquist-Majda Boundary
Conditions ............................................ 127
7.3. Maxwell's Equations in Flat Space ..................... 132
8. Kernels, Chains, and Evolution Operators ................... 137
8.1. A Convolution Calculus with Operator-Valued Kernels ... 138
8.2. Chains ................................................ 146
8.3. Juxtaposition of Chains ............................... 147
8.4. Finitely Generated Chains ............................. 148
8.5. Evolution Operators ................................... 149
8.6. Stable Families of Generators ......................... 155
9. The Linear Evolution Equation .............................. 165
10. Examples of Linear Evolution Equations .................... 177
10.1. Scalar Fields in the Gravitational Field of a
Spherical Black Hole ................................. 178
10.2. Non-Autonomous Linear Hermitian Hyperbolic Systems ... 199
11. The Quasi-Linear Evolution Equation ....................... 215
12. Examples of Quasi-Linear Evolution Equations .............. 235
12.1. A Generalized Inviscid Burgers' Equation ............ 235
12.2. Quasi-Linear Hermitian Hyperbolic Systems ........... 246
13. Appendix .................................................. 265
References .................................................... 269
Index of Notation ............................................. 279
Index of Terminology .......................................... 281
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