Note: Front Cover -- Quantum Chaos -- Copyright Page -- Table of Contents -- Preface -- Chapter 1. Hyperbolic Structure in Classical Chaos -- 1. Introduction. -- 2. Transverse homoclinic orbits -- 3. Anti-integrable limit -- 4. Interlude -- 5. Anosov systems -- 6. Nonzero Lyapunov exponents -- 7. Flux -- 8. Summary -- A guide to the literature -- APPENDIX: Mathematical notation -- Chapter 2. A New Paradigm in Quantum Chaos: Aubry's Theory of Equilibrium States for the Adiabatic Holstein Model -- 1. Introduction -- 2. The model -- 3. The anti-integrable limit t = 0 -- 4. Small hopping -- 5. Explicit estimates -- 6. Properties -- 7. Extensions -- APPENDIX A: Notation and basic mathematical results -- APPENDIX B: Comments on the proof of [1] -- APPENDIX C: Solution of a recurrence inequality -- APPENDIX D: Variation of the electronic energy with configuration u -- REFERENCES -- Chapter 3. Periodic-Orbit Theory -- 1. Introduction -- 2. Semi-classical periodic-orbit theory -- 3. Organizing chaos -- 4. Symmetries -- 5. The three-disk system -- 6. Classical periodic-orbit theory -- 7. Matrix elements -- 8. Final remarks -- REFERENCES -- Chapter 4. The Semi-Classical Helium Atom. -- 1. Introduction -- 2. Classical motion in helium -- 3. Semi-classical quantization -- 4. Adiabatic vs. chaotic motion -- 5. Summary and conclusions -- REFERENCES -- Chapter 5. The Riemann Zeta-Function and Quantum Chaology -- 1. The Riemann zeta-function -- 2. The functional equation -- 3. The staircase of zeros -- 4. The quantum chaology connection -- 5. Convergence properties of periodic-orbit formulae -- 6. Riemann-Siegel resummation -- 7. The pair correlation of the zeros -- APPENDIX: Probabilistic number theory and the pairwise distribution of the primes -- REFERENCES -- Chapter 6. Quantum Localization -- 1. Introduction -- 2. The kicked rotor -- 3. Anderson localization. , 4. The mapping of the kicked-rotor problem on the Anderson model -- 5. Adiabatic localization -- 6. Measures and manifestations of localization -- 7. Summary -- 8. Related problems -- REFERENCES -- Chapter 7. Dynamical Localization in the Hydrogen Atom -- 1. Introduction -- 2. Classical dynamics -- 3. One-dimensional model -- 4. Kepler map -- 5. Photonic localization -- 6. Derealization -- 7. Quantization of the Kepler map and the scattering problem -- 9. Conclusion -- REFERENCES -- Chapter 8. Dynamical Localization, Dissipation and Noise -- 1. Introduction -- 2. Dissipative quantum dynamics -- 3. Dynamical localization in the dissipative kicked-rotor model -- 4. Rydberg atoms in a noisy waveguide -- REFERENCES -- Chapter 9. Statistics of Quasi-Energy Spectrum. -- 1. Introduction -- 2. Some time-dependent models with classical chaos -- 3. The kicked-rotator model -- 4. General properties of the quasi-energy spectrum and quantum resonance -- 5. Uncorrelated statistics of quasi-energy -- 6. Maximal statistical properties of quasi-energy spectra -- 7. Intermediate statistics caused by the localization -- REFERENCES -- Chapter 10. Scattering and Resonances: Classical and Quantum Dynamics. -- 1. Introduction -- 2. Classical scattering and chaotic repellers -- 3. Semi-classical quantum scattering -- 4. Conclusions -- REFERENCES -- Chapter 11. Exotic Fractals and Atomic Decay. -- 1. Introduction -- 2. What is the one-dimensional kicked hydrogen atom (1DKH)? -- 3. Why consider the 1DKH? -- 4. Equations of motion -- 5. The mapping -- 6. The never-come-back property -- 7. The Jacobian -- 8. Eigenvalues -- 9. Hyperbolicity -- 10. The tent map, a model for ionization -- 11. Power law decay of the 1DKH -- 12. Exotic (scale-broken) fractals -- 13. Summary -- REFERENCES -- Chapter 12. Quantum Chaotic Scattering and Microwave Experiments. -- 1. Introduction. , 2. The model -- 3. Semi-classical theory -- 4. S-matrix element energy correlations -- 5. S-matrix correlations under general perturbations -- 6. The Wigner time delay -- 7. Absorption -- 8. Conclusions -- REFERENCES.

Note: Title Page -- Indice -- Preface -- Gruppo fotografico dei partecipanti al Corso -- Classical and quantum zeta-functions and periodic orbits theory -- Introduction -- Escape rates, zeta-functions and determinants -- The strange repeller and its escape rate -- Topological aspects, cycle expansions -- Convergence properties of cycle expansion -- Spectra and spaces -- General structure of cycle expansion -- Curvatures -- Symbolic dynamics tricky points -- Dynamical averages in terms of zeta-functions -- Averages and generating functions -- A model average calculation -- Dynamical averages and deterministic diffusion -- A cycle expansion for the diffusion constant -- A piecewise linear example -- The inclusion of marginal stability -- One-dimensional intermittent maps -- Anomalous diffusion -- Zeta-functions without periodic orbits -- Semiclassics: Gutzwiller trace formula -- Density of states and Green's functions -- The WKB ansatz -- Van Vleck formula -- The semiclassical Green's function -- The short-time contribution to G -- Gutzwiller trace formula -- Quantum determinants -- Appendix A -- Manipulations on Van Vleck's determinant -- Appendix B -- From action to monodromy matrix -- Spectral twinkling -- Introduction -- Examples of singularity-dominated strong fluctuations -- Smells in random winds, and the sex life of moths -- van Hove singularities and kin -- Twinkling starlight -- Spectral twinkling for integrable systems: superpoisson fluctuations -- Spectral twinkling for mixed systems -- Chaos, dissipation and quantal Brownian motion -- Introduction -- Definition of the problem -- Restricted versions of the problem -- History" of the problem -- Fluctuations: intensity and correlation time -- Fluctuations: time-dependent Hamiltonian -- Actual, parametric and reduced energy changes -- The sudden and the adiabatic approximations. , Ballistic and diffusive energy spreading -- Energy spreading and dissipation -- Application to the "piston" example -- The route to stochastic behavior -- The transition probability kernel -- Limitations on quantal-classical correspondence (QCC) -- The parametric evolution of P(n|m) -- The time evolution of P_t(n|m) -- Linear response theory -- Actual and parametric dynamics -- Perturbation theory -- The over-simplified RMT picture -- The perturbative core-tail spreading profile -- An improved perturbation theory -- Consequences of the improved perturbative treatment -- The quantum-mechanical sudden approximation -- The quantum-mechanical adiabatic approximation -- Classical Brownian motion -- The DLD Hamiltonian -- The white-noise approximation (WNA) -- Consequences of the WNA -- The reduced propagator -- Master equation -- Brownian motion and dephasing -- The open question -- Quantum chaos in extended systems: Spreading wave packets and avoided band crossings -- Introduction -- Avoided band crossings -- Modeling avoided band crossings -- A perturbation calculation for the three-band model -- A real example -- Dynamics of quantum systems with fractal spectrum -- The spectral function -- Uniform scaling -- Multiscaling -- A brief introduction to random matrices -- Introduction -- Gaussian ensembles of Hermitian matrices -- Integration measure -- Eigenvalue distribution -- Eigenvector distribution -- Spacing distributions -- Why universality of spectral fluctuations? -- Wigner's semicircle law -- Dyson's circular ensembles -- Two-point correlator and form factor -- CUE averages of symmetric functions -- CUE form factor and pair correlator -- How to process a single spectrum into a two-point correlator -- Gaussian statistics and independence of finite-time traces -- Ergodicity of the form factor -- Poisson ensemble -- Non-Gaussian ensembles. , Universality of spectral fluctuations -- Autocorrelations of secular determinants -- The semiclassical tool in mesoscopic physics -- Introduction -- Quantum transport through classically chaotic cavities -- Chaotic scattering -- The scattering approach to the conductance -- Quantum interference in ballistic cavities -- Semiclassical transmission amplitudes -- Transmission coefficients and average values -- Conductance fluctuations -- Weak localization in the ballistic regime -- Scattering and integrability in quantum transport -- Direct trajectories -- Scattering through a rectangular cavity -- Semiclassical transmission amplitudes for square cavities -- Mean conductance in a square cavity -- Conductance fluctuations in a square cavity -- Circular billiards, diffraction and tunneling -- Experiments on ballistic transport and other aspects of the theory -- Conductance fluctuations and weak localization in ballistic microstructures -- Semiclassics vs. random matrix theory -- Mixed dynamics -- Semiclassical approach to bulk conductivity -- Orbital magnetism in clean systems -- Thermodynamic formalism -- Semiclassical treatment of susceptibilities -- Square billiards -- Integrable vs. chaotic behavior -- Semiclassical approach to weak disorder -- Disorder models -- Single-particle Green function -- Two-particle Green function -- Fixed-size impurity average of the magnetic susceptibility -- Combined impurity and energy average of the susceptibility -- Relation to experiment and other theories -- Electron-electron interactions in the ballistic regime -- Screened Coulomb interaction in two dimensions -- Thermodynamics and semiclassics of small interacting systems -- First-order perturbation, diagonal and non-diagonal contributions -- Higher-order terms -- Conclusions. , Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach -- Introduction -- Introduction to the supersymmetry method and application to RMT -- Green's function approach -- Supermathematics -- Average DOS from supersymmetry -- Level correlations -- Comments and generalizations -- Structure of the saddle point manifold -- Gaussian ensembles of different symmetry -- Ensembles with non-Gaussian distributions of matrix elements -- Random banded matrices -- Parametric level statistics -- Level statistics in a disordered sample: Diffusive sigma-model -- Derivation of the diffusive sigma-model -- Reduction to the 0D sigma-model: Universal limit -- Deviations from universality -- Perturbation theory -- Deviations from universality at omega < -- < -- E_c -- Stationary-point method -- Spectral characteristics related to R_2(s) -- Spectral form factor -- Level number variance -- Eigenfunction statistics -- Random matrix theory -- Eigenfunction statistics in terms of the supersymmetric sigma-model -- Quasi-one-dimensional geometry -- Exact solution of the sigma-model -- Short wire -- Long wire -- Metallic regime (arbitrary d) -- 2D: Weak multifractality of eigenfunctions -- Spatial correlations of eigenfunction amplitudes -- Zero-mode approximation -- Quasi-1D geometry -- Metallic regime (arbitrary d) -- Anomalously localized states and long-time relaxation -- Quasi-1D geometry -- 2D geometry -- Random matrix model -- Supersymmetry approach to the quantum chaos -- Introduction: What we have learned from the diffusive problem -- Ballistic sigma-model -- Heuristic arguments -- Ballistic sigma-model from disorder averaging -- sigma-model from energy averaging -- Non-universal corrections and statistical noise -- Problem of repetitions -- Billiard with diffuse surface scattering -- Level statistics. , The level number variance -- Eigenfunction statistics -- sigma-model for the kicked rotor -- Concluding remarks -- Experimental study of quantum chaos with cold atoms -- Introduction -- A two-level atom in a standing-wave potential -- Experimental method -- Kicked rotor -- Current and future directions -- Quantum stochasticity and the many-body problem -- Introduction -- Symmetries and the semiclassical approximation -- Motivation -- Pedestrian approach -- Symmetry arguments -- Invariant chaotic manifolds and collective motion -- Invariant manifolds in interacting many-body systems -- Quantum case -- Classical model for the giant dipole resonance -- Localization in Hilbert space -- Anderson localization -- Localization in polyatomic molecules -- Fock-space localization in quantum dots -- Fock-space localization in self-bound fermionic many-body systems -- Spectral statistics and periodic orbits -- Introduction -- Generalities -- Trace formulas -- Random matrix theory -- Correlation functions -- Diagonal approximation -- Criterion of applicability of diagonal approximation -- Beyond the diagonal approximation -- The Hardy-Littlewood conjecture -- Arithmetical systems -- Construction of the density of states from a finite number of periodic orbits -- Off-diagonal terms for the Riemann zeta-function -- Off-diagonal contribution for dynamical systems -- Random matrix universality -- Riemann-Siegel form of density of states -- Conclusion -- Quantum chaos and thermalization for interacting particles -- Introduction -- Two-body random interaction model -- Description of the model -- Many-body Hamiltonian -- Structure of the Hamiltonian matrix -- Correlations in off-diagonal matrix elements -- Density of states and spectrum statistics -- Structure of exact eigenstates -- Strength function -- Analytical solution for the LDOS. , Non-statistical properties of the TBRI-model.