Random Matrices and the Statistical Theory of Energy Levels

Preface1 / Introduction 1.1. The Need to Study Random Matrices 1.2. A Summary of Statistical Facts about Nuclear Energy Levels 1.3. Definition of a Suitable Function for the Study of Level Correlations 1.4. Wigner Surmise 1.5. Electromagnetic Properties of Small Metallic Particles2 / Gaussian Ensembles. The Joint Probability Density Function of the Matrix Elements 2.1. Preliminaries 2.2. Time-Reversal Invariance 2.3. Gaussian Orthogonal Ensemble 2.4. Gaussian Symplectic Ensemble 2.5. Gaussian Unitary Ensemble 2.6. Joint Probability Density Function for Matrix Elements3 / Gaussian Ensembles. The Joint Probability Density Function of the Eigenvalues 3.1. Orthogonal Ensemble 3.2. Symplectic Ensemble 3.3. Unitary Ensemble4 / Gaussian Ensembles 4.1. The Partition Function 4.2. The Asymptotic Formula for the Level Density5 / The Gaussian Orthogonal Ensemble 5.1. General Remarks 5.2. The Method of Integration over Alternate Variables 5.3. The Normalization Constant 5.4. One- and Two-Level Correlation Functions 5.5. Level Spacings 5.6. Bounds for the Distribution Function F(t)6 / The Gaussian Unitary and Symplectic Ensembles 6.1. The Gaussian Unitary Ensemble 6.2. Gaussian Symplectic Ensemble7 / Brownian Motion Model 7.1. Stationary Ensembles 7.2. Nonstationary Ensembles 7.3. Some Ensemble Averages8 / Circular Ensembles 8.1. General Remarks 8.2. The Orthogonal Ensemble 8.3. Symplectic Ensemble 8.4. Unitary Ensemble 8.5. The Joint Probability Density Function for the Eigenvalues9 / Circular Ensembles. Correlation Functions, Spacing Distribution, etc. 9.1. Orthogonal Ensemble 9.2. Symplectic Ensemble, ß = 4 9.3. Unitary Ensemble, ß = 2 9.4. Browning Motion Model10 / Circular Ensembles. Thermodynamics 10.1. General Remarks 10.2. The Partition Function 10.3. Thermodynamic Quantities 10.4. Statistical Interpretation of U and C 10.5. Continuum Model for the Spacing Distribution11 / The Orthogonal Circular Ensemble. Wigners Method 11.1. General Remarks 11.2. The Method of Integration 11.3. Spacing Distribution12 / Matrices with Gaussian Element Densities But with No Unitary or Hermitian Condition Imposed 12.1. Complex Matrices 12.2. Quaternion Matrices 12.3. Real Matrices13 / Gaussian Ensembles. Level Density in the Tail of the Semicircle Text14 / Bordered Matrices 14.1. Random Linear Chain 14.2. Bordered Matrices15 / Invariance Hypothesis and Matrix Element Correlations Text16 / The Joint Probability Density Functions for Two Nearby Spacings 16.1. Integrations 16.2. An Integral Equation with a Boundary Condition and A2m(θ, α) 16.3. The Limit of A2m(θ, α) 16.4. Power Series Expansion and Numerical Results 16.5. The Distribution of Spacings between Next-Nearest Neighbors17 / Restricted Trace Ensembles. Ensembles Related to the Classical Orthogonal Polynomials 17.1. Fixed Trace Ensemble 17.2. Bounded Trace Ensembles 17.3. Matrix Ensembles and Classical Orthogonal PolynomialsAppendices A.1. Proof of Equation (2.52) A.2. Counting the Dimensions of TßG and T'ßG (Chapter 3) and of Tßc and T'ßc (Chapter 8) A.3. Two Proofs of Equation (4.5) for the Case N = 3 A.4. The Minimum Value of W, Equation (4.6) A.5. Proof of Equation (4.15) A.6. Proof of Equations (5.4), (5.51), and (5.52) A.7. Proof of Equation (5.20). Expansion of a Pfaffian along Its Principal Pseudodiagonal A.8. The Limit of ΣN0-1 φi2(x) A.9. The Limits of ΣN0-1 φ1(x) φi(y) etc A.10. The Fourier Transforms of the Two-Point Cluster Functions A.11. Proof of Equations (5.84) and (9.39) A.12. Various Probability Distribution and Probability Density Functions A.13. Some Applications of Gram's Result A.14. Power-Series Expansion of Im(θ) A.15. Proof of the Inequalities (5.130) A.16. The Confluent Alternant A.17. Proof of Equations (8.12) and (8.28) A.18. Proof of Equation (11.29) A.19. Wilson's Proof of Equation (10.11) A.20. Proof That the Second Term in Equation (7.28) Drops Out on Summation A.21. Proof of the Inequality (10.5) A.22. The Probability Density of the Spacings Resulting from a Random Superposition of n Unrelated Sequences of Energy Levels A.23. Some Properties Connected with Symmetric and Antisymmetric Unitary Matrices A.24. Evaluation of the Interval (12.9) for Complex Matrices A.25. A Few Remarks about the Eigenvalues of a Quaternion Real Matrix and Its Diagonalization A.26. Evaluation of the Integral (12.46) A.27. The Proof of Equation (12.70) A.28. The Case of Random Real Matrices A.29. The Density of Eigenvalues of a Random Matrix Whose Elements All Have the Same Mean Square Value A.30. Values of the Functions B(x1 , x2) (Table A.30.1), and P(x1 , x2) (Table A.30.2) A.31. Proof of Equations (6.25") and (16.69)ReferencesIndex