Theory of Foundations of Real Variables(哈佛数学系的Shlomo Sternberg教授实变函数书)

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Theory of Foundations of Real Variables(哈佛数学系的Shlomo Sternberg教授实变函数书)
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Contents
1 The topology of metric spaces 13
1.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Completeness and completion. . . . . . . . . . . . . . . . . . . . . 16
1.3 Normed vector spaces and Banach spaces. . . . . . . . . . . . . . 17
1.4 Compactness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Total Boundedness. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Separability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Second Countability. . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Conclusion of the proof of Theorem 1.5.1. . . . . . . . . . . . . . 20
1.9 Dini’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.10 The Lebesgue outer measure of an interval is its length. . . . . . 21
1.11 Zorn’s lemma and the axiom of choice. . . . . . . . . . . . . . . . 23
1.12 The Baire category theorem. . . . . . . . . . . . . . . . . . . . . 24
1.13 Tychonoff’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.14 Urysohn’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.15 The Stone-Weierstrass theorem. . . . . . . . . . . . . . . . . . . . 27
1.16 Machado’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.17 The Hahn-Banach theorem. . . . . . . . . . . . . . . . . . . . . . 32
1.18 The Uniform Boundedness Principle. . . . . . . . . . . . . . . . . 35
2 Hilbert Spaces and Compact operators. 37
2.1 Hilbert space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Scalar products. . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.2 The Cauchy-Schwartz inequality. . . . . . . . . . . . . . . 38
2.1.3 The triangle inequality . . . . . . . . . . . . . . . . . . . . 39
2.1.4 Hilbert and pre-Hilbert spaces. . . . . . . . . . . . . . . . 40
2.1.5 The Pythagorean theorem. . . . . . . . . . . . . . . . . . 41
2.1.6 The theorem of Apollonius. . . . . . . . . . . . . . . . . . 42
2.1.7 The theorem of Jordan and von Neumann. . . . . . . . . 42
2.1.8 Orthogonal projection. . . . . . . . . . . . . . . . . . . . . 45
2.1.9 The Riesz representation theorem. . . . . . . . . . . . . . 47
2.1.10 What is L2(T)? . . . . . . . . . . . . . . . . . . . . . . . . 48
2.1.11 Projection onto a direct sum. . . . . . . . . . . . . . . . . 49
2.1.12 Projection onto a finite dimensional subspace. . . . . . . . 49
5
6 CONTENTS
2.1.13 Bessel’s inequality. . . . . . . . . . . . . . . . . . . . . . . 49
2.1.14 Parseval’s equation. . . . . . . . . . . . . . . . . . . . . . 50
2.1.15 Orthonormal bases. . . . . . . . . . . . . . . . . . . . . . 50
2.2 Self-adjoint transformations. . . . . . . . . . . . . . . . . . . . . . 51
2.2.1 Non-negative self-adjoint transformations. . . . . . . . . . 52
2.3 Compact self-adjoint transformations. . . . . . . . . . . . . . . . 54
2.4 Fourier’s Fourier series. . . . . . . . . . . . . . . . . . . . . . . . 57
2.4.1 Proof by integration by parts. . . . . . . . . . . . . . . . . 57
2.4.2 Relation to the operator d
dx . . . . . . . . . . . . . . . . . . 60
2.4.3 G°arding’s inequality, special case. . . . . . . . . . . . . . . 62
2.5 The Heisenberg uncertainty principle. . . . . . . . . . . . . . . . 64
2.6 The Sobolev Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.7 G°arding’s inequality. . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.8 Consequences of G°arding’s inequality. . . . . . . . . . . . . . . . 76
2.9 Extension of the basic lemmas to manifolds. . . . . . . . . . . . . 79
2.10 Example: Hodge theory. . . . . . . . . . . . . . . . . . . . . . . . 80
2.11 The resolvent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3 The Fourier Transform. 85
3.1 Conventions, especially about 2. . . . . . . . . . . . . . . . . . . 85
3.2 Convolution goes to multiplication. . . . . . . . . . . . . . . . . . 86
3.3 Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4 Fourier transform of a Gaussian is a Gaussian. . . . . . . . . . . 86
3.5 The multiplication formula. . . . . . . . . . . . . . . . . . . . . . 88
3.6 The inversion formula. . . . . . . . . . . . . . . . . . . . . . . . . 88
3.7 Plancherel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.8 The Poisson summation formula. . . . . . . . . . . . . . . . . . . 89
3.9 The Shannon sampling theorem. . . . . . . . . . . . . . . . . . . 90
3.10 The Heisenberg Uncertainty Principle. . . . . . . . . . . . . . . . 91
3.11 Tempered distributions. . . . . . . . . . . . . . . . . . . . . . . . 92
3.11.1 Examples of Fourier transforms of elements of S0. . . . . . 93
4 Measure theory. 95
4.1 Lebesgue outer measure. . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Lebesgue inner measure. . . . . . . . . . . . . . . . . . . . . . . . 98
4.3 Lebesgue’s definition of measurability. . . . . . . . . . . . . . . . 98
4.4 Caratheodory’s definition of measurability. . . . . . . . . . . . . . 102
4.5 Countable additivity. . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6 -fields, measures, and outer measures. . . . . . . . . . . . . . . . 108
4.7 Constructing outer measures, Method I. . . . . . . . . . . . . . . 109
4.7.1 A pathological example. . . . . . . . . . . . . . . . . . . . 110
4.7.2 Metric outer measures. . . . . . . . . . . . . . . . . . . . . 111
4.8 Constructing outer measures, Method II. . . . . . . . . . . . . . . 113
4.8.1 An example. . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.9 Hausdorff measure. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.10 Hausdorff dimension. . . . . . . . . . . . . . . . . . . . . . . . . . 117
CONTENTS 7
4.11 Push forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.12 The Hausdorff dimension of fractals . . . . . . . . . . . . . . . . 119
4.12.1 Similarity dimension. . . . . . . . . . . . . . . . . . . . . . 119
4.12.2 The string model. . . . . . . . . . . . . . . . . . . . . . . 122
4.13 The Hausdorff metric and Hutchinson’s theorem. . . . . . . . . . 124
4.14 Affine examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.14.1 The classical Cantor set. . . . . . . . . . . . . . . . . . . . 126
4.14.2 The Sierpinski Gasket . . . . . . . . . . . . . . . . . . . . 128
4.14.3 Moran’s theorem . . . . . . . . . . . . . . . . . . . . . . . 129
5 The Lebesgue integral. 133
5.1 Real valued measurable functions. . . . . . . . . . . . . . . . . . 134
5.2 The integral of a non-negative function. . . . . . . . . . . . . . . 134
5.3 Fatou’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4 The monotone convergence theorem. . . . . . . . . . . . . . . . . 140
5.5 The space L1(X,R). . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6 The dominated convergence theorem. . . . . . . . . . . . . . . . . 143
5.7 Riemann integrability. . . . . . . . . . . . . . . . . . . . . . . . . 144
5.8 The Beppo - Levi theorem. . . . . . . . . . . . . . . . . . . . . . 145
5.9 L1 is complete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.10 Dense subsets of L1(R,R). . . . . . . . . . . . . . . . . . . . . . 147
5.11 The Riemann-Lebesgue Lemma. . . . . . . . . . . . . . . . . . . 148
5.11.1 The Cantor-Lebesgue theorem. . . . . . . . . . . . . . . . 150
5.12 Fubini’s theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.12.1 Product -fields. . . . . . . . . . . . . . . . . . . . . . . . 151
5.12.2 -systems and -systems. . . . . . . . . . . . . . . . . . . 152
5.12.3 The monotone class theorem. . . . . . . . . . . . . . . . . 153
5.12.4 Fubini for finite measures and bounded functions. . . . . 154
5.12.5 Extensions to unbounded functions and to -finite measures.156
6 The Daniell integral. 157
6.1 The Daniell Integral . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2 Monotone class theorems. . . . . . . . . . . . . . . . . . . . . . . 160
6.3 Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.4 H¨older, Minkowski , Lp and Lq. . . . . . . . . . . . . . . . . . . . 163
6.5 k · k1 is the essential sup norm. . . . . . . . . . . . . . . . . . . . 166
6.6 The Radon-Nikodym Theorem. . . . . . . . . . . . . . . . . . . . 167
6.7 The dual space of Lp. . . . . . . . . . . . . . . . . . . . . . . . . 170
6.7.1 The variations of a bounded functional. . . . . . . . . . . 171
6.7.2 Duality of Lp and Lq when μ(S) < 1. . . . . . . . . . . . 172
6.7.3 The case where μ(S) = 1. . . . . . . . . . . . . . . . . . 173
6.8 Integration on locally compact Hausdorff spaces. . . . . . . . . . 175
6.8.1 Riesz representation theorems. . . . . . . . . . . . . . . . 175
6.8.2 Fubini’s theorem. . . . . . . . . . . . . . . . . . . . . . . . 176
6.9 The Riesz representation theorem redux. . . . . . . . . . . . . . . 177
6.9.1 Statement of the theorem. . . . . . . . . . . . . . . . . . . 177
8 CONTENTS
6.9.2 Propositions in topology. . . . . . . . . . . . . . . . . . . 178
6.9.3 Proof of the uniqueness of the μ restricted to B(X). . . . 180
6.10 Existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.10.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.10.2 Measurability of the Borel sets. . . . . . . . . . . . . . . . 182
6.10.3 Compact sets have finite measure. . . . . . . . . . . . . . 183
6.10.4 Interior regularity. . . . . . . . . . . . . . . . . . . . . . . 183
6.10.5 Conclusion of the proof. . . . . . . . . . . . . . . . . . . . 184
7 Wiener measure, Brownian motion and white noise. 187
7.1 Wiener measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.1.1 The Big Path Space. . . . . . . . . . . . . . . . . . . . . . 187
7.1.2 The heat equation. . . . . . . . . . . . . . . . . . . . . . . 189
7.1.3 Paths are continuous with probability one. . . . . . . . . 190
7.1.4 Embedding in S0. . . . . . . . . . . . . . . . . . . . . . . . 194
7.2 Stochastic processes and generalized stochastic processes. . . . . 195
7.3 Gaussian measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.3.1 Generalities about expectation and variance. . . . . . . . 196
7.3.2 Gaussian measures and their variances. . . . . . . . . . . 198
7.3.3 The variance of a Gaussian with density. . . . . . . . . . . 199
7.3.4 The variance of Brownian motion. . . . . . . . . . . . . . 200
7.4 The derivative of Brownian motion is white noise. . . . . . . . . . 202
8 Haar measure. 205
8.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.1.1 Rn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.1.2 Discrete groups. . . . . . . . . . . . . . . . . . . . . . . . 206
8.1.3 Lie groups. . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.2 Topological facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.3 Construction of the Haar integral. . . . . . . . . . . . . . . . . . 212
8.4 Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.5 μ(G) < 1 if and only if G is compact. . . . . . . . . . . . . . . . 218
8.6 The group algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.7 The involution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.7.1 The modular function. . . . . . . . . . . . . . . . . . . . . 220
8.7.2 Definition of the involution. . . . . . . . . . . . . . . . . . 222
8.7.3 Relation to convolution. . . . . . . . . . . . . . . . . . . . 223
8.7.4 Banach algebras with involutions. . . . . . . . . . . . . . 223
8.8 The algebra of finite measures. . . . . . . . . . . . . . . . . . . . 223
8.8.1 Algebras and coalgebras. . . . . . . . . . . . . . . . . . . . 224
8.9 Invariant and relatively invariant measures on homogeneous spaces.225
CONTENTS 9
9 Banach algebras and the spectral theorem. 231
9.1 Maximal ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.1.1 Existence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9.1.2 The maximal spectrum of a ring. . . . . . . . . . . . . . . 232
9.1.3 Maximal ideals in a commutative algebra. . . . . . . . . . 233
9.1.4 Maximal ideals in the ring of continuous functions. . . . . 234
9.2 Normed algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
9.3 The Gelfand representation. . . . . . . . . . . . . . . . . . . . . . 236
9.3.1 Invertible elements in a Banach algebra form an open set. 238
9.3.2 The Gelfand representation for commutative Banach algebras.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.3.3 The spectral radius. . . . . . . . . . . . . . . . . . . . . . 241
9.3.4 The generalized Wiener theorem. . . . . . . . . . . . . . . 242
9.4 Self-adjoint algebras. . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.4.1 An important generalization. . . . . . . . . . . . . . . . . 247
9.4.2 An important application. . . . . . . . . . . . . . . . . . . 248
9.5 The Spectral Theorem for Bounded Normal Operators, Functional
Calculus Form. . . . . . . . . . . . . . . . . . . . . . . . . 249
9.5.1 Statement of the theorem. . . . . . . . . . . . . . . . . . . 250
9.5.2 SpecB(T) = SpecA(T). . . . . . . . . . . . . . . . . . . . . 251
9.5.3 A direct proof of the spectral theorem. . . . . . . . . . . . 253
10 The spectral theorem. 255
10.1 Resolutions of the identity. . . . . . . . . . . . . . . . . . . . . . 256
10.2 The spectral theorem for bounded normal operators. . . . . . . . 261
10.3 Stone’s formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
10.4 Unbounded operators. . . . . . . . . . . . . . . . . . . . . . . . . 262
10.5 Operators and their domains. . . . . . . . . . . . . . . . . . . . . 263
10.6 The adjoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
10.7 Self-adjoint operators. . . . . . . . . . . . . . . . . . . . . . . . . 265
10.8 The resolvent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.9 The multiplication operator form of the spectral theorem. . . . . 268
10.9.1 Cyclic vectors. . . . . . . . . . . . . . . . . . . . . . . . . 269
10.9.2 The general case. . . . . . . . . . . . . . . . . . . . . . . . 271
10.9.3 The spectral theorem for unbounded self-adjoint operators,
multiplication operator form. . . . . . . . . . . . . . 271
10.9.4 The functional calculus. . . . . . . . . . . . . . . . . . . . 273
10.9.5 Resolutions of the identity. . . . . . . . . . . . . . . . . . 274
10.10The Riesz-Dunford calculus. . . . . . . . . . . . . . . . . . . . . . 276
10.11Lorch’s proof of the spectral theorem. . . . . . . . . . . . . . . . 279
10.11.1Positive operators. . . . . . . . . . . . . . . . . . . . . . . 279
10.11.2 The point spectrum. . . . . . . . . . . . . . . . . . . . . . 281
10.11.3Partition into pure types. . . . . . . . . . . . . . . . . . . 282
10.11.4 Completion of the proof. . . . . . . . . . . . . . . . . . . . 283
10.12Characterizing operators with purely continuous spectrum. . . . 287
10.13Appendix. The closed graph theorem. . . . . . . . . . . . . . . . 288
10 CONTENTS
11 Stone’s theorem 291
11.1 von Neumann’s Cayley transform. . . . . . . . . . . . . . . . . . 292
11.1.1 An elementary example. . . . . . . . . . . . . . . . . . . . 297
11.2 Equibounded semi-groups on a Frechet space. . . . . . . . . . . . 299
11.2.1 The infinitesimal generator. . . . . . . . . . . . . . . . . . 299
11.3 The differential equation . . . . . . . . . . . . . . . . . . . . . . . 301
11.3.1 The resolvent. . . . . . . . . . . . . . . . . . . . . . . . . . 303
11.3.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
11.4 The power series expansion of the exponential. . . . . . . . . . . 309
11.5 The Hille Yosida theorem. . . . . . . . . . . . . . . . . . . . . . . 310
11.6 Contraction semigroups. . . . . . . . . . . . . . . . . . . . . . . . 313
11.6.1 Dissipation and contraction. . . . . . . . . . . . . . . . . . 314
11.6.2 A special case: exp(t(B − I)) with kBk  1. . . . . . . . . 316
11.7 Convergence of semigroups. . . . . . . . . . . . . . . . . . . . . . 317
11.8 The Trotter product formula. . . . . . . . . . . . . . . . . . . . . 320
11.8.1 Lie’s formula. . . . . . . . . . . . . . . . . . . . . . . . . . 320
11.8.2 Chernoff’s theorem. . . . . . . . . . . . . . . . . . . . . . 321
11.8.3 The product formula. . . . . . . . . . . . . . . . . . . . . 322
11.8.4 Commutators. . . . . . . . . . . . . . . . . . . . . . . . . 323
11.8.5 The Kato-Rellich theorem. . . . . . . . . . . . . . . . . . 323
11.8.6 Feynman path integrals. . . . . . . . . . . . . . . . . . . . 324
11.9 The Feynman-Kac formula. . . . . . . . . . . . . . . . . . . . . . 326
11.10The free Hamiltonian and the Yukawa potential. . . . . . . . . . 328
11.10.1 The Yukawa potential and the resolvent. . . . . . . . . . . 329
11.10.2 The time evolution of the free Hamiltonian. . . . . . . . . 331
12 More about the spectral theorem 333
12.1 Bound states and scattering states. . . . . . . . . . . . . . . . . . 333
12.1.1 Schwartzschild’s theorem. . . . . . . . . . . . . . . . . . . 333
12.1.2 The mean ergodic theorem . . . . . . . . . . . . . . . . . 335
12.1.3 General considerations. . . . . . . . . . . . . . . . . . . . 336
12.1.4 Using the mean ergodic theorem. . . . . . . . . . . . . . . 339
12.1.5 The Amrein-Georgescu theorem. . . . . . . . . . . . . . . 340
12.1.6 Kato potentials. . . . . . . . . . . . . . . . . . . . . . . . 341
12.1.7 Applying the Kato-Rellich method. . . . . . . . . . . . . . 343
12.1.8 Using the inequality (12.7). . . . . . . . . . . . . . . . . . 344
12.1.9 Ruelle’s theorem. . . . . . . . . . . . . . . . . . . . . . . . 345
12.2 Non-negative operators and quadratic forms. . . . . . . . . . . . 345
12.2.1 Fractional powers of a non-negative self-adjoint operator. 345
12.2.2 Quadratic forms. . . . . . . . . . . . . . . . . . . . . . . . 346
12.2.3 Lower semi-continuous functions. . . . . . . . . . . . . . . 347
12.2.4 The main theorem about quadratic forms. . . . . . . . . . 348
12.2.5 Extensions and cores. . . . . . . . . . . . . . . . . . . . . 350
12.2.6 The Friedrichs extension. . . . . . . . . . . . . . . . . . . 350
12.3 Dirichlet boundary conditions. . . . . . . . . . . . . . . . . . . . 351
12.3.1 The Sobolev spaces W1,2(
) and W1,2
0 (
). . . . . . . . . 352

CONTENTS 11
12.3.2 Generalizing the domain and the coefficients. . . . . . . . 354
12.3.3 A Sobolev version of Rademacher’s theorem. . . . . . . . 355
12.4 Rayleigh-Ritz and its applications. . . . . . . . . . . . . . . . . . 357
12.4.1 The discrete spectrum and the essential spectrum. . . . . 357
12.4.2 Characterizing the discrete spectrum. . . . . . . . . . . . 357
12.4.3 Characterizing the essential spectrum . . . . . . . . . . . 358
12.4.4 Operators with empty essential spectrum. . . . . . . . . . 358
12.4.5 A characterization of compact operators. . . . . . . . . . 360
12.4.6 The variational method. . . . . . . . . . . . . . . . . . . . 360
12.4.7 Variations on the variational formula. . . . . . . . . . . . 362
12.4.8 The secular equation. . . . . . . . . . . . . . . . . . . . . 364
12.5 The Dirichlet problem for bounded domains. . . . . . . . . . . . 365
12.6 Valence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
12.6.1 Two dimensional examples. . . . . . . . . . . . . . . . . . 367
12.6.2 H¨uckel theory of hydrocarbons. . . . . . . . . . . . . . . . 368
12.7 Davies’s proof of the spectral theorem . . . . . . . . . . . . . . . 368
12.7.1 Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
12.7.2 Slowly decreasing functions. . . . . . . . . . . . . . . . . . 369
12.7.3 Stokes’ formula in the plane. . . . . . . . . . . . . . . . . 370
12.7.4 Almost holomorphic extensions. . . . . . . . . . . . . . . . 371
12.7.5 The Heffler-Sj¨ostrand formula. . . . . . . . . . . . . . . . 371
12.7.6 A formula for the resolvent. . . . . . . . . . . . . . . . . . 373
12.7.7 The functional calculus. . . . . . . . . . . . . . . . . . . . 374
12.7.8 Resolvent invariant subspaces. . . . . . . . . . . . . . . . 376
12.7.9 Cyclic subspaces. . . . . . . . . . . . . . . . . . . . . . . . 377
12.7.10 The spectral representation. . . . . . . . . . . . . . . . . . 380
13 Scattering theory. 383
13.1 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
13.1.1 Translation - truncation. . . . . . . . . . . . . . . . . . . . 383
13.1.2 Incoming representations. . . . . . . . . . . . . . . . . . . 384
13.1.3 Scattering residue. . . . . . . . . . . . . . . . . . . . . . . 386
13.2 Breit-Wigner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
13.3 The representation theorem for strongly contractive semi-groups. 388
13.4 The Sinai representation theorem. . . . . . . . . . . . . . . . . . 390
13.5 The Stone - von Neumann theorem. . . . . . . . . . . . . . . . . 392

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