
Long Time Behaviour of Classical and Quantum Systems : Proceedings of the Bologna Aptex International Conference, Bologna, Italy, 13-17 September 1999.
Title:
Long Time Behaviour of Classical and Quantum Systems : Proceedings of the Bologna Aptex International Conference, Bologna, Italy, 13-17 September 1999.
Author:
Graffi, Sandro.
ISBN:
9789812794598
Personal Author:
Physical Description:
1 online resource (299 pages)
Series:
Series on Concrete and Applicable Mathematics ; v.1
Series on Concrete and Applicable Mathematics
Contents:
CONTENTS -- Foreword -- List of Participants -- Return to Equilibrium in Classical and Quantum Systems -- 1. FIRST LECTURE -- 2. SECOND LECTURE -- 3. THIRD LECTURE -- 4. FOURTH LECTURE -- Quantum Resonances and Trapped Trajectories -- 1 Introduction -- 2 Definitions and first results -- 3 FBI transform and upper bounds on the density of resonances -- 4 Trace formulae and lower bounds on the density of resonances -- 5 Resonances and a non-linear Schrodinger equation -- Return to Thermal Equilibrium in Quantum Statistical Mechanics -- 1. INTRODUCTION -- 2. EQUILIBRIUM STATES AND DYNAMICS OF FINITE SYSTEMS -- 3. EQUILIBRIUM STATES AND DYNAMICS OF INFINITE SYSTEMS -- 4. EXAMPLE: FINITE QUANTUM SYSTEM -- 5. EXAMPLE: FREE QUANTIZED ELECTROMAGNETIC FIELD IN THE ARAKI-WOODS REPRESENTATION -- 6. EXAMPLE: CONFINED ELECTRON AND FREE QUANTIZED ELECTROMAGNETIC FIELD - NONINTERACTING -- 7. CONFINED ELECTRON COUPLED TO THE QUANTIZED RADIATION FIELD - INTERACTING CASE -- Small Oscillations in Some Nonlinear PDE's -- 1 Introduction -- 2 The finite dimensional case -- 3 The infinite dimensional case: some known results -- 4 A proof of Lyapunov center theorem: the finite dimensional case -- 5 The resonant case -- 6 A proof of the Lyapunov center theorem: the infinite dimensional case -- 7 On the verification of the property y-NR -- 8 Applications -- The Semi-Classical Van-Vleck Formula. Application to the Aharonov-Bohm Effect -- 1 Introduction -- 2 Coherent states and quantum propagator -- 3 Semi-classical approximation for the propagator -- 4 The time-dependent Aharonov-Bohm Effect -- Fractal Dimensions and Quantum Evolution Associated with Sparse Potential Jacobi Matrices -- 1 Introduction -- 2 The sparse barrier model and main results -- 3 Pictures of quantum motion within sparse barriers -- 4 Proof of Theorem 2 -- 5 Proof of Theorem 3.
6 Conclusions -- Infinite Step Billiards -- 1 Introduction -- 2 The model and the results -- 3 Outline of the proofs -- Semiclassical Expansion for the Thermodynamic Limit of the Ground State Energy of Kac's Operator -- 1 Introduction -- 2 One-parameter families of weighted standard functions -- 3 WKB constructions -- 4 A formal asymptotic expansion -- 5 Estimates for the thermodynamic limit -- Asymptotics of Scattering Poles for Two Strictly Convex Obstacles -- 1. INTRODUCTION -- 2. METHOD OF THE PROOF -- 3. EXPRESSION OF BROKEN RAYS CONVERGING TO THE PERIODIC RAY -- 4. SOLUTIONS OF FUNCTION EQUATIONS -- 5. TAYLOR EXPANSION OF Tn(s+t o+r) -- Parabolic Dynamical Systems and Inducing -- 1 Preliminaires -- 2 Parabolic rational maps -- QFT for Scalar Particles in External Fields on Riemannian Manifolds -- 1 Introduction -- 2 Invariant wave equations on Riemannnian manifolds -- 3 Classical S-matrix -- 4 Feynman's scattering amplitude -- 5 Solvability of the equation AF = A + iAP_AF -- 6 Quantum field theory in external forces -- 7 Hilbert-Schmidt property on Riemannian manifolds -- 8 Massless case -- Existence and Born-Oppenheimer Asymptotics of the Total Scattering Cross-Section in Ion-Atom Collisions -- I Introduction -- II Notation assumptions and main results -- III Existence of the total scattering cross-section -- IV Born-Oppenheimer approximation of oa -- A Expansion of the potentials -- On Asymptotic Perturbation Theory for Quantum Mechanics -- 1 Introduction -- 2 Almost invariant subspaces. General theory -- 3 Perturbative construction of almost invariant subspaces -- 4 Asymptotic estimations -- 5 Magnetic Schrodinger operators: gauge invariant perturbation theory -- Destruction of the Beating Effect in a Periodically Driven Double-Well -- 1 Introduction. -- 2 Assumptions -- 3 Main result -- 4 The explicit model.
5 Splitting vanishing and localization -- Berezin-Toeplitz Quantization and Berezin Transform -- 1. INTRODUCTION -- 2. THE SET-UP -- 3. THE SEMI-CLASSICAL BEHAVIOUR AND THE BEREZIN-TOEPLITZ DEFORMATION QUANTIZATION -- 4. THE TOEPLITZ STRUCTURE -- 5. BEREZIN SYMBOLS AND THE BEREZIN TRANSFORM -- 6. APPLICATIONS OF THE ASYMPTOTICS OF THE BEREZIN TRANSFORM -- 7. SOME COMMENTS ON THE DYNAMICS -- APPENDIX A. MORE DETAILS ON THE ASYMPTOTIC EXPANSION.
Abstract:
This book is centered on the two minicourses conducted by C Liverani (Rome) and J Sjoestrand (Paris) on the return to equilibrium in classical statistical mechanics and the location of quantum resonances via semiclassical analysis, respectively. The other contributions cover related topics of classical and quantum mechanics, such as scattering theory, classical and quantum statistical mechanics, dynamical localization, quantum chaos, ergodic theory and KAM techniques. Contents: Return to Equilibrium in Classical and Quantum Systems (C Liverani); Quantum Resonances and Trapped Trajectories (J Sjöstrand); Return to Thermal Equilibrium in Quantum Statistical Mechanics (V Bach); Small Oscillations in Some Nonlinear PDE's (D Bambusi & S Paleari); The Semi-Classical Van-Vleck Formula. Application to the Aharonov-Bohm Effect (J-M Bily & D Robert); Fractal Dimensions and Quantum Evolution Associated with Sparse Potential Jacobi Matrices (J-M Combes & G Mantica.); Infinite Step Billiards (M D Esposti); Semiclassical Expansion for the Thermodynamic Limit of the Ground State Energy of Kac's Operator (B Helffer & T Ramond); Asymptotics of Scattering Poles for Two Strictly Convex Obstacles (M Ikawa); Parabolic Dynamical Systems and Inducing (S Isola); QFT for Scalar Particles in External Fields on Riemannian Manifolds (H Isozaki); Existence and Born-Oppenheimer Asymptotics of the Total Scattering Cross-Section in Ion-Atom Collisions (T Jecko et al.); On Asymptotic Perturbation Theory for Quantum Mechanics (G Nenciu); Destruction of the Beating Effect in a Periodically Driven Double-Well (A Sacchetti); Berezin-Toeplitz Quantization and Berezin Transform (M Schlichenmaier). Readership: Researchers, academics and graduate students in mechanics, mathematical physics and mathematics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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