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Foundations of Superposition Theory vol

De
314 pages
Foundations of Superposition Theory vol. 1 Superposition Algebra in the Space of Tempered Distributions and Applications to Economics and Physics David Carfı Edizioni Il Gabbiano 2010

  • fourier transforms

  • esummable families

  • superposition

  • slinear hulls

  • relationships among

  • frechet spaces

  • topological supplements

  • dirac basis

  • dieudonne-schwartz theorem

  • sclosed subsets


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Foundations of Superposition Theory
vol. 1
Superposition Algebra in the Space of Tempered Distributions
and Applications to Economics and Physics
David Carf
Edizioni Il Gabbiano 20102Contents
I Introductions and preliminaries 11
0.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1 Preliminaries 17
1.1 Topological homomorphisms . . . . . . . . . . . . . . . . . . 18
1.2 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Topological supplements . . . . . . . . . . . . . . . . . . . . . 20
1.4 Right and Left inverses . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Homomorphisms among Frechet spaces . . . . . . . . . . . 21
1.6 Dieudonne-Schwartz theorem . . . . . . . . . . . . . . . . . . 22
1.7 Banach-Steinhaus in barreled spaces . . . . . . . . . . . . . 23
1.8 Tempered distributions . . . . . . . . . . . . . . . . . . . . . 24
1.8.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . 24
1.8.2 Tempered distributions . . . . . . . . . . . . . . . . . 25
1.9 Fourier transforms onS . . . . . . . . . . . . . . . . . . . . 25n
01.10 F onS . . . . . . . . . . . . . . . . . . . . 26n
II Superpositions 27
2 Summable families 29
2.1 Families of distributions . . . . . . . . . . . . . . . . . . . . . 29
S2.2 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
S2.3 Family generated by an operator . . . . . . . . . . . . . . . 31
S2.4 The operator generated by an family . . . . . . . . . . . . 32
S2.5 Characterizations of families . . . . . . . . . . . . . . . . . 33
2.6 Characterization of transposability . . . . . . . . . . . . . . 36
D2.7 of families (*) . . . . . . . . . . . . . . . 37
E E2.8 Families and summable families . . . . . . . . . . . . . . 37
3 Superpositions 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 The wonderful Dirac basis . . . . . . . . . . . . . . . 41
3.1.2 A dangerous expression . . . . . . . . . . . . . . . . . 42
34 CONTENTS
3.1.3 Toward a possible solution . . . . . . . . . . . . . . . 43
3.1.4 Inadequacy of convolutions . . . . . . . . . . . . . . . 44
3.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 45
S 03.2 Superpositions of families inS . . . . . . . . . . . . . . . 45n
3.3 An alternative de nition of superposition . . . . . . . . . . 46
3.4 Superpositions of an E-family (*) . . . . . . . . . . . . . . . 48
3.5 Algebraic properties of superpositions . . . . . . . . . . . . 49
3.5.1 Bilinearity of superposition operator . . . . . . . . . 49
3.5.2 Selection property of the Dirac distributions . . . . 50
S3.5.3 Linear combination of an family . . . . . . . . . . . 51
S4 Linearity 53
4.1 Continuity of superposition operators . . . . . . . . . . . . 53
4.2 Superposition operator of a distribution . . . . . . . . . . . 54
S4.3 Linearity of superpositions . . . . . . . . . . . . . . . . . . . 56
4.4 Generalized distributive laws (*) . . . . . . . . . . . . . . . . 58
S5 Families inS 61n
5.1 Families inS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61n
S5.2 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
S5.3 Transpose of an Family . . . . . . . . . . . . . . . . . . . . . 64
S5.4 Operator of an family . . . . . . . . . . . . . . . . . . . . . . 65
S5.5 Continuity of operators of families . . . . . . . . . . . . . 66
6 Superpositions inS 69n
S6.1 Linear combinations . . . . . . . . . . . . . . . . . . . . . . . 69
S6.2 Superposition operator of families . . . . . . . . . . . . . 70
S6.3 Summability of families . . . . . . . . . . . . . . . . . . . . 71
6.4 Transpose family . . . . . . . . . . . . . . . . . . . . . . . . . . 72
S6.5 Linear functional . . . . . . . . . . . . . . . . . . . . . . . . . 72
S 06.6 Superpositions of families inS . . . . . . . . . . . . . . . 73n
S6.7 Linear superpositions of operators . . . . . . . . . . . . . . 75
7 First applications 77
7.1 The Fourier expansion theorem . . . . . . . . . . . . . . . . 77
7.2 Convolution as superpositions . . . . . . . . . . . . . . . . . 80
7.3 Some expressions of Dirac Calculus . . . . . . . . . . . . . . 82
7.3.1 The expansion of a vector in the Dirac basis . . . . 83
7.3.2 Fourier expansions and the momentum operator . 84
7.4 Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 85
SIII Linear Algebra and Geometry 89
S S8 Linear hulls of families 91
S8.1 Linear hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91CONTENTS 5
S8.2 Algebraic properties of linear hulls . . . . . . . . . . . . . 92
S8.3 Systems of generators . . . . . . . . . . . . . . . . . . . . . . 94
8.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
S8.4 Topological properties of linear hulls . . . . . . . . . . . . 96
S8.5 Closedness of linear hulls . . . . . . . . . . . . . . . . . . . 97
S8.5.1 Examples of systems of generators . . . . . . . . . 98
S8.6 Kernel of an family . . . . . . . . . . . . . . . . . . . . . . . 99
8.6.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . 100
S8.7 Linear hull of a subset . . . . . . . . . . . . . . . . . . . . . 101
9 Bases 103
S9.1 Linear independence . . . . . . . . . . . . . . . . . . . . . . 103
S9.2 Topology and linear independence . . . . . . . . . . . . . . 104
9.3 Uniqueness of representation . . . . . . . . . . . . . . . . . . 106
S9.4 Characterizations of linear independence . . . . . . . . . 106
S9.5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
S9.6 Algebraic characterizations of bases . . . . . . . . . . . . 108
9.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
S9.7 Totality of bases . . . . . . . . . . . . . . . . . . . . . . . . . 109
S9.8 Topological characterizations of bases . . . . . . . . . . . 110
S9.9 Equivalent families . . . . . . . . . . . . . . . . . . . . . . . . 110
S10 Closedness 113
S10.1 Closed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . 113
S10.2 hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3 Relationships among di erent hulls . . . . . . . . . . . . . . 115
10.3.1 Relationships among linear hulls . . . . . . . . . . . 116
S10.3.2 Closure of subspaces . . . . . . . . . . . . . . . . . . 117
S10.3.3 among linear and closed hulls 117
S10.4 Unions of closed sets . . . . . . . . . . . . . . . . . . . . . . 119
S10.5 Closed linear hull of families . . . . . . . . . . . . . . . . . 119
S10.6 Closedness and topology . . . . . . . . . . . . . . . . . . . . 120
S11 Connectedness 123
S D11.1 Connected and connected sets . . . . . . . . . . . . . . . 123
S 011.1.1 Families inS containing a given distribution . . . 124n
S 011.1.2 F in starshaped subsets of S . . . . . . . . 124n
S 011.1.3 Connected subsets of S . . . . . . . . . . . . . . . . 126n
D 1L11.2 Closed sets (*) . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.2.1 Preliminaries on the spaceD 1 . . . . . . . . . . . . . 127L
11.3 Sums of series as superpositions . . . . . . . . . . . . . . . . 1296 CONTENTS
SIV Linear operators 133
S12 Linear operators 135
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
S12.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
S 012.3 Op de ned on S . . . . . . . . . . . . . . . . . . . . 137n
S 012.4 Characterization of operators onS . . . . . . . . . . . . . 138n
S 012.5 Linear operators onS . . . . . . . . . . . . . . . . . . . . . 139n
S12.6 Examples of linear operators . . . . . . . . . . . . . . . . . 140
S12.6.1 The superposition operator of an family . . . . . . 140
12.6.2 Transpose operators . . . . . . . . . . . . . . . . . . . 141
S12.7 Characterization of linear operators . . . . . . . . . . . . 143
S13 Applications of linear operators 145
S13.1 Bases of subspaces . . . . . . . . . . . . . . . . . . . . . . . . 145
S13.2 of closed subspaces . . . . . . . . . . . . . . . . . . . 146
S13.3 Bases of barreled . . . . . . . . . . . . . . . . . . 148
S13.4 Superpositions of linear operators . . . . . . . . . . . . . 148
S S13.5 Linear operators and bases . . . . . . . . . . . . . . . . . . 150
S13.6 Invertibility of linear operators . . . . . . . . . . . . . . . . 151
S13.7 Linear operators on subspaces . . . . . . . . . . . . . . . . 152
S13.8 Compositions of linear operators . . . . . . . . . . . . . . . 153
S14 Homomorphisms 155
S14.1 . . . . . . . . . . . . . . . . . . . . . . . . . 156
S14.2 Injective linear homomorphisms . . . . . . . . . . . . . . . 157
S14.3 Surjective linear . . . . . . . . . . . . . . 158
S14.4 Stable families . . . . . . . . . . . . . . . . . . . . . . . . . . 159
S S14.5 Linear operators and closedness . . . . . . . . . . . . . . 160
S S14.6 Homomorphism and . . . . . . . . . . . . . . 162
S14.7 Invertibility of linear homomorphism . . . . . . . . . . . 162
S14.8 Left inverse of homomorphisms . . . . . . . . . . . 163
S15 Green’s families 165
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
15.1.1 Green’s functions . . . . . . . . . . . . . . . . . . . . . 165
15.1.2 Motivations for Green’s families . . . . . . . . . . . . 166
015.2 Green’s families inS . . . . . . . . . . . . . . . . . . . . . . . 167n
S15.3 families . . . . . . . . . . . . . . . . . . . . . . . . . . 168
15.4 Application to linear equations . . . . . . . . . . . . . . . . . 169
15.5 Interpretation of the solution . . . . . . . . . . . . . . . . . . 170
15.6 Characterization of Green’s families . . . . . . . . . . . . . 172
15.7 Existence of Green’s families . . . . . . . . . . . . . . . . . . 173
15.8 Translation invariance and convolutions . . . . . . . . . . . 174
15.8.1 Example: the Laplacian . . . . . . . . . . . . . . . . . 175
S S15.9 Green’s families relative to bases . . . . . . . . . . . . . . 176CONTENTS 7
V Representations 179
S16 Coordinates 181
16.1 Systems of coordinates . . . . . . . . . . . . . . . . . . . . . . 181
16.2 Coordinate operators . . . . . . . . . . . . . . . . . . . . . . . 182
16.3 Basic properties of coordinate operators . . . . . . . . . . . 183
16.3.1 Linear properties . . . . . . . . . . . . . . . . . . . . . 183
S16.3.2 prop . . . . . . . . . . . . . . . . . . . . . 184
16.3.3 Topological properties . . . . . . . . . . . . . . . . . . 186
S16.4 Coordinate operators of stable families . . . . . . . . . . . 187
S16.5 Hulls of stable families . . . . . . . . . . . . . . . . . . . . . 188
S16.6 Linearity of the coordinate operator . . . . . . . . . . . . 189
S17 Applications of Coordinates 191
S17.1 Invertibility of homomorphism . . . . . . . . . . . . . . . . 191
17.2 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
17.3 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . 194
17.4 Superpositions respect to operators . . . . . . . . . . . . . . 195
17.5 Resolution of the identity . . . . . . . . . . . . . . . . . . . . 197
S18 Matrices 199
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
18.2 Product of families . . . . . . . . . . . . . . . . . . . . . . . . 200
18.3 Transpose product of families . . . . . . . . . . . . . . . . . 202
18.4 Invertible families . . . . . . . . . . . . . . . . . . . . . . . . . 203
18.5 Coordinates and invertible families . . . . . . . . . . . . . . 205
19 Representations in QM 211
S19.1 Representation of endomorphisms . . . . . . . . . . . . . . 211
19.2 of vector states . . . . . . . . . . . . . . . . 213
19.2.1 First examples . . . . . . . . . . . . . . . . . . . . . . . 213
19.3 The representation correspondence . . . . . . . . . . . . . . 214
19.4 Representation ofS-linear operators . . . . . . . . . . . . . 216
19.5 Classic examples in QM . . . . . . . . . . . . . . . . . . . . . 217
S19.5.1 Matrix representations . . . . . . . . . . . . . . . . . 217
19.5.2 Operatortation . . . . . . . . . . . . . . . . . 218
SVI Spectral theory 221
m 020 Multiplicative operators inS(R ;S ) 223n
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
OM20.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
20.2.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
(n)
20.2.2 Bounded sets inO . . . . . . . . . . . . . . . . . . . 225M
(n)
OM20.2.3 Multiplications by functions . . . . . . . . . . . . 2258 CONTENTS
S20.2.4 Family of the multiplication operator M . . . . . 226f
OM20.3 Product inL(S ;S ) by functions . . . . . . . . . . . . . 226n m
(m)
20.3.1 The ringO . . . . . . . . . . . . . . . . . . . . . . . . 227M
20.3.2 The moduleL(S ;S ) . . . . . . . . . . . . . . . . . . . 228n m
S OM20.4 Product of families by functions . . . . . . . . . . . . . 228
O SM20.5 Functions and basis . . . . . . . . . . . . . . . . . . . 230
O SM20.6 Invertible functions and basis . . . . . . . . . . . . . . . 232
21 Spectral expansions 235
S21.1 Spectral expansions . . . . . . . . . . . . . . . . . . . . . . . 235
S S21.2 Expansions and linear equations . . . . . . . . . . . . . . 237
21.3 Existence of Green families . . . . . . . . . . . . . . . . . . . 239
21.4 Superpositions inO . . . . . . . . . . . . . . . . . . . . . . . 242M
S22 Diagonalizable operators 245
S 022.1 operatorsS . . . . . . . . . . . . . . . . . . 245n
S22.2 op onS . . . . . . . . . . . . . . . . 247n
S22.3 Algebra of diagonalizable operators . . . . . . . . . . . . . 248
22.4 Building some observables of QM . . . . . . . . . . . . . . . 252
22.4.1 The position operator in one dimension . . . . . . . 252
22.4.2 The p op in three dimensions . . . . . 252
22.4.3 The momentum operator . . . . . . . . . . . . . . . . 253
22.4.4 The kinetic energy in dimension 1 . . . . . . . . . . 253
22.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
22.5.1 Observables with a singular spectrum . . . . . . . . 255
22.5.2 The relativistic energy . . . . . . . . . . . . . . . . . . 255
23 Spectrum 257
23.1 Supports and vanishing-laws . . . . . . . . . . . . . . . . . . 257
23.2 Structure of the eigenspectrum . . . . . . . . . . . . . . . . 259
23.3 of the spectrum . . . . . . . . . . . . . . . . . . . . 260
24 Functional Calculus 263
24.1 A vanishing lemma . . . . . . . . . . . . . . . . . . . . . . . . 263
S24.2 Transformable diagonalizable operators . . . . . . . . . . 265
S24.3 Functions of operators . . . . . . . . . . . . 268
24.4 Compatibility with the exponential . . . . . . . . . . . . . . 270
SVII Linear Dynamics 273
25 The Schr odinger equation 275
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
25.2 Di erentiable curve . . . . . . . . . . . . . . . . . . . . . . . . 275
25.2.1 Di erentiable curves in topological vector spaces . 275
025.2.2 curves in S . . . . . . . . . . . . . . . 277nCONTENTS 9
25.2.3 Di erentiable curves in O (*) . . . . . . . . . . . . 278M
25.2.4 Pointwise di erentiable curve in O . . . . . . . . . 279M
025.3 The Schr odinger’s equation in S . . . . . . . . . . . . . . . 281n
25.3.1 Solutions of the eigen-representation . . . . . . . . . 281
025.3.2 The Schr odinger’s equation in S . . . . . . . . . . . 282n
25.3.3 The evolution group . . . . . . . . . . . . . . . . . . . 285
025.3.4 The abstract Heat equation onS . . . . . . . . . . . 285n
S26 Diagonalizable equations 287
26.1 Superpositions with respect to curves . . . . . . . . . . . . 287
26.2 Solution of the eigenrepresentations . . . . . . . . . . . . . 288
26.3 The eigenrepresentations . . . . . . . . . . . . . . . . . . . . 290
26.4 Solutions in functional form . . . . . . . . . . . . . . . . . . . 291
27 Feynman propagators 293
27.1 Propagators as family-valued functions . . . . . . . . . . . 293
27.2 Evolution operators . . . . . . . . . . . . . . . . . . . . . . . . 294
27.3 Operatorial propagators . . . . . . . . . . . . . . . . . . . . . 297
27.4 Feynman propagator of a free particle . . . . . . . . . . . . 298
028 Evolutions in the spaceS 301n
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
m 028.2 Integral of function fromR toS . . . . . . . . . . . . . . 302n
m 028.3 In of functions of R intoL(S ) . . . . . . . . . . . . 305n
28.4 Curves ofS-linear operators . . . . . . . . . . . . . . . . . . 306
28.5 The Dyson Formula . . . . . . . . . . . . . . . . . . . . . . . . 31010 CONTENTS

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