1 Introduction
References
2 Time Reversal and Unitary Symmetries
2.1 Autonomous Classical Flows
2.2 Spinless Quanta
2.3 Spin- 1/2 Quanta
2.4 Hamiltonians Without T Invariance
2.5 T Invariant Hamiltonians, T2 = 1
2.6 Kramers' Degeneracy
2.7 Kramers' Degeneracy and Geometric Symmetries
2.8 Kramers' Degeneracy Without Geometric Symmetries
2.9 Nonconventional Time Reversal
2.10 Stroboscopic Maps for Periodically Driven Systems
2.11 Time Reversal for Maps
2.12 Canonical Transformations for Floquet Operators
2.13 Beyond Dyson's Threefold Way
2.13.1 Normal-Superconducting Hybrid Structures.
2.13.2 Systems with Chiral Symmetry
2.14 Problems
References
3 Level Repulsion
3. l Preliminaries
3.2 Symmetric Versus Nonsymmetric H or F
3.3 Kramers' Degeneracy
3.4 Universality Classes of Level Repulsion
3.5 Nonstandard Symmetry Classes
3.6 Experimental Observation of Level Repulsion
3.7 Problems
References
4 Random-Matrix Theory
4.1 Preliminaries
4.2 Ganssian Ensembles of Hermitian Matrices
4.3 Eigenvalue Distributions for Dyson's Ensembles
4.4 Eigenvalue Distributions for Nonstandard Symmetry Classes
4.5 Level Spacing Distributions
4.6 Invariance of the Integration Measure
4.7 Average Level Density
4.8 Unfolding Spectra
4.9 Eigenvector Distributions
4.9. l Single-Vector Density
4.9.2 Joint Density of Eigenvectors
4.10 Ergodicity of the Level Density
4.11 Dyson's Circular Ensembles
4.12 Asymptotic Level Spacing Distributions
4.13 Determinants as Gaussian Grassmann Integrals
4.14 Two-Point Correlations of the Level Density
4.14.1 Two-Point Correlator and Form Factor
4.14.2 Form Factor for the Poissonian Ensemble
4.14.3 Form Factor for the CUE
4.14.4 Form Factor for the COE
4.14.5 Form Factor for the CSE
4.15 Newton's Relations
4.15.1 Traces Versus Secular Coefficients
4.15.2 Solving Newton's Relations
4.16 Selfinversiveness and Riemann-Siegel Lookalike
4.17 Higher Correlations of the Level Density
4.17.1 Correlation and Cumulant Functions
4.17.2 Ergodicity of the Two-Point Correlator
4.17.3 Ergodicity of the Form Factor
4.17.4 Joint Density of Traces of Large CUE Matrices
4.18 Correlations of Secular Coefficients
4.19 Fidelity of Kicked Tops to Random-Matrix Theory
4.20 Problems
References
5 Level Clustering
5.1 Preliminaries
5.2 Invariant Tori of Classically Integrable Systems
5.3 Einstein-Brillouin-Keller Approximation
5.4 Level Crossings for Integrable Systems
5.5 Poissonian Level Sequences
5.6 Superposition of Independent Spectra
5.7 Periodic Orbits and the Semiclassical Density of Levels
5.8 Level Density Fluctuations for Integrable Systems
5.9 Exponential Spacing Distribution for Integrable Systems
5.10 Equivalence of Different Unfoldings
5.11 Problems
References
6 Level Dynamics
6.1 Preliminaries
6.2 Fictitious Particles (Pechukas-Yukawa Gas)
6.3 Conservation Laws
6.4 Intermultiplet Crossings
6.5 Level Dynamics for Classically Integrable Dynamics
6.6 Two-Body Collisions
6.7 Ergodicity of Level Dynamics and Universality
of Spectral Fluctuations
6.7.1 Ergodicity
6.7.2 Collision Time
6.7.3 Universality
6.8 Equilibrium Statistics
6.9 Random-Matrix Theory as Equilibrium Statistical Mechanics
6.9.1 General Strategy
6.9.2 A Typical Coordinate Integral
6.9.3 Influence of a Typical Constant of the Motion
6.9.4 The General Coordinate Integral
6.9.5 Concluding Remarks
6.10 Dynamics of Rescaled Energy Levels
6.11 Level Curvature Statistics
6.12 Level Velocity Statistics
6.13 Dyson's Brownian-Motion Model
6.14 Local and Global Equilibrium in Spectra
6.15 Problems
References
7 Quantum Localization
7.1 Preliminaries
7.2 Localization in Anderson's Hopping Model
7.3 The Kicked Rotator as a Variant of Anderson's Model
7.4 Lloyd's Model
7.5 The Classical Diffusion Constant as the Quantum Localization
Length
7.6 Absence of Localization for the Kicked Top
7.7 The Rotator as a Limiting Case of the Top
7.8 Problems
References
8 Dissipative Systems
8.1 Preliminaries
8.2 Hamiltonian Embeddings
8.3 Time-Scale Separation for Probabilities and Coherences
8.4 Dissipative Death of Quantum Recurrences
8.5 Complex Energies and Quasi-Energies
8.6 Different Degrees of Level Repulsion for Regular
and Chaotic Motion
8.7 Poissonian Random Process in the Plane
8.8 Ginibre's Ensemble of Random Matrices
8.8.1 Normalizing the Joint Density
8.8.2 The Density of Eigenvalues
8.8.3 The Reduced Joint Densities
8.8.4 The Spacing Distribution
8.9 General Properties of Generators
8.10 Universality of Cubic Level Repulsion
8.10.1 Antiunitary Symmetries
8.10.2 Microreversibility
8.11 Dissipation of Quantum Localization
8.11.1 Zaslavsky's Map
8.11.2 Damped Rotator
8.11.3 Destruction of Localization
8.12 Problems
References
9 Classical Hamiltonian Chaos
9.1 Preliminaries
9.2 Phase Space, Hamilton's Equations and All That
9.3 Action as a Generating Function
9.4 Linearized Flow and Its Jacobian Matrix
9.5 Liouville Picture
9.6 Symplectic Structure
9.7 Lyapunov Exponents
9.8 Stretching Factors and Local Stretching Rates
9.9 Poincar6 Map
9.10 Stroboscopic Maps of Periodically Driven Systems
9.11 Varieties of Chaos
9.12 The Sum Rule of Hannay and Ozorio de Almeida
9.12.1 Maps
9.12.2 Flows
9.13 Propagator and Zeta Function
9.14 Exponential Stability of the Boundary Value Problem ..
9.15 Sieber-Richter Self-Encounter and Partner Orbit
9.15.1 Non-technical Discussion
9.15.2 Quantitative Discussion of 2-Encounters
9.16 l-Encounters and Orbit Bunches
9.17 Densities of Arbitrary Encounter Sets
9.18 Problems
References
10 Semiclassical Roles for Classical Orbits
10.1 Preliminaries
10.2 Van Vleck Propagator
10.2.1 Maps
10.2.2 Flows
10.3 Gutzwiller's Trace Formula
10.3.1 Maps
10.3.2 Flows
10.3.3 Weyrs Law
10.3.4 Limits of Validity and Outlook
10.4 Lagrangian Manifolds and Maslov Theory
10.4.1 Lagrangian Manifolds
10.4.2 Elements of Maslov Theory
10.4.3 Maslov Indices as Winding Numbers
10.5 Riemann-Siegel Look-Alike
10.6 Spectral Two-Point Correlator
10.6.1 Real and Complex Correlator
10.6.2 Local Energy Average
10.6.3 Generating Function
10.6.4 Periodic-Orbit Representation
10.7 Diagonal Approximation
10.7.1 Unitary Class
10.7.2 Orthogonal Class
10.8 Off-Diagonal Contributions, Unitary Symmetry Class
10.8.1 Structures of Pseudo-Orbit Quadruplets
10.8.2 Diagrammatic Rules
10.8.3 Example of Structure Contributions: A Single
2-encounter
10.8.4 Cancellation of all Encounter Contributions
for the Unitary Class
10.9 Semiclassical Construction of a Sigma Model,Unitary Symmetry Class
10.9.1 Matrix Elements for Ports and Contraction Lines for Links
10.9.2 Wick's Theorem and Link Summation
10.9.3 Signs
10.9.4 Proof of Contraction Rules, Unitary Case
10.9.5 Emergence of a Sigma Model
10.10 Semiclassical Construction of a Sigma Model,
Orthogonal Symmetry Class
10.10.1 Structures
10.10.2 Leading-Order Contributions
10.10.3 Symbols for Ports and Contraction Lines for Links
10.10.4 Gauss and Wick
10.10.5 Signs
10.10.6 Proof of Contraction Rules, Orthogonal Case
10.10.7 Sigma Model
10.11 Outlook
10.12 Mixed Phase Space
10.13 Problems
References
11 Superanalysis for Random-Matrix Theory
11.1 Preliminaries
11.2 Semicircle Law for the Gaussian Unitary Ensemble
11.2.1 The Green Function and Its Average
11.2.2 The GUE Average
11.2.3 Doing the Superintegral
11.2.4 Two Remaining Saddle-Point Integrals
11.3 Superalgebra
11.3.1 Motivation and Generators of Grassmann Algebras
11.3.2 Supervectors, Supermatrices
11.3.3 Superdeterminants
11.3.4 Complex Scalar Product, Hermitian
and Unitary Supermatrices
11.3.5 Diagonalizing Supermatrices
11.4 Superintegrals
11.4.1 Some Bookkeeping for Ordinary Gaussian Integrals
11.4.2 Recalling Grassmann Integrals
11.4.3 Gaussian Superintegrals
11.4.4 Some Properties of General Superintegrals
11.4.5 Integrals over Supermatrices,
Parisi-Sourlas-Efetov-Wegner Theorem
11.5 The Semicircle Law Revisited
11.6 The Two-Point Function of the Gaussian Unitary Ensemble
11.6.1 The Generating Function
11.6.2 Unitary Versus Hyperbolic Symmetry
11.6.3 Efetov's Nonlinear Sigma Model
11.6.4 Implementing the Zero-Dimensional Sigma Model
11.6.5 Integration Measure of the Nonlinear Sigma Model
11.6.6 Back to the Generating Function
11.6.7 Rational Parametrization of the Sigma Model
11.6.8 High-Energy Asymptotics
11.7 Universality of Spectral Fluctuations:
Non-Gaussian Ensembles
11.7.1 Delta Functions of Grassmann Variables
11.7.2 Generating Function
11.8 Universal Spectral Fluctuations of Sparse Matrices
11.9 Thick Wires, Banded Random Matrices,
One-Dimensional Sigma Model
11.9.1 Banded Matrices Modelling Thick Wires
11.9.2 Inverse Participation Ratio and Localization Length
11.9.3 One-Dimensional Nonlinear Sigma Model
11.9.4 Implementing the One-Dimensional Sigma Model
11.10 Problems
References
Index
