Random matrices and the statistical theory of energy levels:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Academic Press
1967
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | 2. Aufl. u.d.T.: Mehta, Madan L.: Random matrices |
| Physical Description: | X, 259 S. |
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| 100 | 1 | |a Mehta, Madan Lal |d 1932-2006 |e Verfasser |0 (DE-588)13927667X |4 aut | |
| 245 | 1 | 0 | |a Random matrices and the statistical theory of energy levels |c M. L. Mehta |
| 264 | 1 | |a New York [u.a.] |b Academic Press |c 1967 | |
| 300 | |a X, 259 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a 2. Aufl. u.d.T.: Mehta, Madan L.: Random matrices | ||
| 650 | 7 | |a Fisica Estatistica (Mec Estatistica) |2 larpcal | |
| 650 | 4 | |a Matrices | |
| 650 | 7 | |a Matrices |2 gtt | |
| 650 | 4 | |a Mécanique statistique | |
| 650 | 7 | |a Spectrometrie |2 gtt | |
| 650 | 4 | |a Énergie, Bande d' (Physique) | |
| 650 | 4 | |a Energy levels (Quantum mechanics) | |
| 650 | 4 | |a Matrices | |
| 650 | 4 | |a Statistical mechanics | |
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Record in the Search Index
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Contents
Preface v
1 / Introduction
1.1. The Need to Study Random Matrices 1
1.2. A Summary of Statistical Facts about Nuclear Energy Levels 5
1.3. Definition of a Suitable Function for the Study of Level Correlations . 8
1.4. Wigner Surmise 10
1.5. Electromagnetic Properties of Small Metallic Particles 12
2 / Gaussian Ensembles. The Joint Probability Density
Function of the Matrix Elements
2.1. Preliminaries 14
2.2. Time Reversal Invariance 15
2.3. Gaussian Orthogonal Ensemble 17
2.4. Gaussian Symplectic Ensemble 18
2.5. Gaussian Unitary Ensemble 23
2.6. Joint Probability Density Function for Matrix Elements 24
3 / Gaussian Ensembles. The Joint Probability Density
Function of the Eigenvalues
3.1. Orthogonal Ensemble 30
3.2. Symplectic Ensemble 33
3.3. Unitary Ensemble 36
4 / Gaussian Ensembles
4.1. The Partition Function 39
4.2. The Asymptotic Formula for the Level Density 43
vii
viii Contents
5 / The Gaussian Orthogonal Ensemble
5.1. General Remarks 47
5.2. The Method of Integration over Alternate Variables 49
5.3. The Normalization Constant 54
5.4. One and Two Level Correlation Functions 57
5.5. Level Spacings 64
5.6. Bounds for the Distribution Function Fit) 70
6/The Gaussian Unitary and Symplectic Ensembles
6.1. The Gaussian Unitary Ensemble 74
6.2. Gaussian Symplectic Ensemble 79
7 / Brownian Motion Model
7.1. Stationary Ensembles 84
7.2. Nonstationary Ensembles 84
7.3. Some Ensemble Averages 90
8 / Circular Ensembles
8.1. General Remarks 93
8.2. The Orthogonal Ensemble 94
8.3. Symplectic Ensemble 96
8.4. Unitary Ensemble 98
8.5. The Joint Probability Density Function for the Eigenvalues 99
9 / Circular Ensembles. Correlation Functions,
Spacing Distribution, etc.
9.1. Orthogonal Ensemble 104
9.2. Symplectic Ensemble. j8 = 4 109
9.3. Unitary Ensemble, ft = 2 112
9.4. Browning Motion Model 114
10/Circular Ensembles. Thermodynamics
10.1. General Remarks 116
10.2. The Partition Function 116
10.3. Thermodynamic Quantities 118
10.4. Statistical Interpretation of U and C 120
10.5. Continuum Model for the Spacing Distribution 122
Contents ix
11 /The Orthogonal Circular Ensemble. Wigner's Method
11.1. General Remarks 128
11.2. The Method of Integration 128
11.3. Spacing Distribution 135
12/Matrices with Gaussian Element Densities But
with No Unitary or Hermitian Condition Imposed
12.1. Complex Matrices 138
12.2. Quaternion Matrices 144
12.3. Real Matrices 150
13/Gaussian Ensembles. Level Density in the Tail of
the Semicircle
Text 151
14/Bordered Matrices
14.1. Random Linear Chain 155
14.2. Bordered Matrices 159
15 / Invariance Hypothesis and Matrix Element Correlations
Text 163
16/The Joint Probability Density Functions for
Two Nearby Spacings
16.1. Integrations 168
16.2. An Integral Equation with a Boundary Condition and A2m($, a) . 171
16.3. The Limit of A2m(8, a) 173
16.4. Power Series Expansion and Numerical Results 176
16.5. The Distribution of Spacings between Next Nearest Neighbors . 178
17 / Restricted Trace Ensembles. Ensembles Related to
the Classical Orthogonal Polynomials
17.1. Fixed Trace Ensemble 180
17.2. Bounded Trace Ensembles 183
17.3. Matrix Ensembles and Classical Orthogonal Polynomials 184
x Contents
Appendices
A.I. Proof of Equation (2.52) 186
A.2. Counting the Dimensions of TpG and T'gG (Chapter 3) and of Tgc
and T'pc (Chapter 8) 186
A.3. Two Proofs of Equation (4.5) for the Case N = 3 188
A.4. The Minimum Value of W, Equation (4.6) •. . . . 191
A.5. Proof of Equation (4.15) 192
A.6. Proof of Equations (5.4), (5.51), and (5.52) 192
A.7. Proof of Equation (5.20). Expansion of a Pfaffian along Its Principal
Pseudodiagonal 194
A.8. The Limit of Ej'"1 p?(.x) 196
A.9. The Limits of E^1 vM 9s(y), etc 197
A. 10. The Fourier Transforms of the Two Point Cluster Functions 199
A.ll. Proof of Equations (5.84) and (9.39) 201
A. 12. Various Probability Distribution and Probability Density Functions . . 202
A. 13. Some Applications of Gram's Result 204
A. 14. Power Series Expansion of Im(B) 206
A.15. Proof of the Inequalities (5.130) 207
A.16. The Confluent Alternant 208
A.17. Proof of Equations (8.12) and (8.28) 209
A.18. Proof of Equation (11.29) 210
A.19. Wilson's Proof of Equation (10.11) 211
A.20. Proof That the Second Term in Equation (7.28) Drops Out on Summation 215
A.21. Proof of the Inequality (10.5) 215
A.22. The Probability Density of the Spacings Resulting from a Random
Superposition of n Unrelated Sequences of Energy Levels 216
A.23. Some Properties Connected with Symmetric and Antisymmetric Unitary
Matrices 219
A.24. Evaluation of the Interval (12.9) for Complex Matrices 226
A.25. A Few Remarks about the Eigenvalues of a Quaternion Real Matrix
and Its Diagonalization 230
A.26. Evaluation of the Integral (12.46) 232
A.27. The Proof of Equation (12.70) 236
A.28. The Case of Random Real Matrices 237
A.29. The Density of Eigenvalues of a Random Matrix Whose Elements
All Have the Same Mean Square Value 238
A.30. Values of the Functions M(xx , x2) (Table A.30.1), and 0'(x1 , x2)
(Table A.30.2) " 240
A.31. Proof of Equations (6.25") and (16.69) 245
References 251
Index 257 |
| any_adam_object | 1 |
| author | Mehta, Madan Lal 1932-2006 |
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| dewey-full | 530.12/3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.12/3 |
| dewey-search | 530.12/3 |
| dewey-sort | 3530.12 13 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik Wirtschaftswissenschaften |
| format | Book |
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| language | English |
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| spelling | Mehta, Madan Lal 1932-2006 Verfasser (DE-588)13927667X aut Random matrices and the statistical theory of energy levels M. L. Mehta New York [u.a.] Academic Press 1967 X, 259 S. txt rdacontent n rdamedia nc rdacarrier 2. Aufl. u.d.T.: Mehta, Madan L.: Random matrices Fisica Estatistica (Mec Estatistica) larpcal Matrices Matrices gtt Mécanique statistique Spectrometrie gtt Énergie, Bande d' (Physique) Energy levels (Quantum mechanics) Statistical mechanics Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001257031&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mehta, Madan Lal 1932-2006 Random matrices and the statistical theory of energy levels Fisica Estatistica (Mec Estatistica) larpcal Matrices Matrices gtt Mécanique statistique Spectrometrie gtt Énergie, Bande d' (Physique) Energy levels (Quantum mechanics) Statistical mechanics Stochastische Matrix (DE-588)4057624-3 gnd |
| subject_GND | (DE-588)4057624-3 |
| title | Random matrices and the statistical theory of energy levels |
| title_auth | Random matrices and the statistical theory of energy levels |
| title_exact_search | Random matrices and the statistical theory of energy levels |
| title_full | Random matrices and the statistical theory of energy levels M. L. Mehta |
| title_fullStr | Random matrices and the statistical theory of energy levels M. L. Mehta |
| title_full_unstemmed | Random matrices and the statistical theory of energy levels M. L. Mehta |
| title_short | Random matrices and the statistical theory of energy levels |
| title_sort | random matrices and the statistical theory of energy levels |
| topic | Fisica Estatistica (Mec Estatistica) larpcal Matrices Matrices gtt Mécanique statistique Spectrometrie gtt Énergie, Bande d' (Physique) Energy levels (Quantum mechanics) Statistical mechanics Stochastische Matrix (DE-588)4057624-3 gnd |
| topic_facet | Fisica Estatistica (Mec Estatistica) Matrices Mécanique statistique Spectrometrie Énergie, Bande d' (Physique) Energy levels (Quantum mechanics) Statistical mechanics Stochastische Matrix |
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