Long Time Behaviour of Classical and Quantum Systems

The following sections are included:

• Foreword

• List of Participants

• CONTENTS

I discuss return to equilibrium both in the classical and quantum setting. In the classical setting the emphasis is on several techniques that have been developed in the last years and that, hopefully, could inspire new approaches in the study of quantum systems. In the quantum setting a very simple, but far from trivial, model is discussed.

The following sections are included:

• Introduction

• Definitions and first results

• FBI transform and upper bounds on the density of resonances

• Trace formulae and lower bounds on the density of resonances

• Resonances and a non-linear Schrödinger equation

• References

We review the research results obtained in collaboration with Jürg Fröhlich and Israel Michael Sigal on relaxation of initial states of a (generalized) spin-boson system at positive temperature to its (unique) thermal equilibrium KMS state [6].

In this paper we study existence of families of periodic orbits close to equilibria of Lagrangian PDE's. We begin by a short survey on the problem, then we illustrate a method recently introduced by the authors to construct small amplitude periodic solutions of some semilinear partial differential equations. As a difference with respect to standard ones it is based on contraction mapping principle, and avoids the use of KAM techinques. Applications to some concrete PDE's are also given. Among others we find small oscillations for a nonlinear plate equation in arbitrary space dimension.

At the very beginning of the quantum theory, Van-Vleck (1928) proposed a nice approximation formula for the integral kernel of the time dependent propagator for the Schrödinger equation. This formula can be deduced from the Feynman path integral by a formal stationary phase argument. After the fondamental works by Hörmander and Maslov on Fourier-integral operators, it became possible to give a rigorous mathematical proof of the Van Vleck formula. We present here a more direct and elementary proof, using propagation of coherent states. We apply this result to give a mathematical proof of the Aharonov-Bohm effect observed on the time dependent propagator. This effect concerns a phase factor depending on the flux of a magnetic field, which can be non trivial, even if the particle never meets the magnetic field.

We study the quantum dynamics generated via Schrödinger equation by sparse-potential Jacobi matrices on l₂(Z₊). Exact bounds for the upper and lower intermittency functions governing the asymptotic growth of moments are derived in terms of the fractal dimensions of the spectral measure. Numerical experiments suggest that these bounds are sharp in the case of very sparse barriers.

In [DDL] and [DDL2] (in collaboration with G. Del Magno e M. Lenci) we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given sequence of non-negative numbers {p_n}_n∈ℕ, such that p_n ↘ 0, there corresponds a table .

The aim of this contribution is to overwiew some of these results. In particular we will focus on the ergodic and topological properties of these non-compact dynamical systesm. We show that generically these systems are ergodic for almost all initial velocities, and the entropy with respect to a wide class of measures is zero.

We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called "transfer operator method". We are interested here in the thermodynamical limit Λ(h) of the ground state energy of this operator. For Kac's spin model, Λ(h) is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation.

Under suitable assumptions, which are satisfied by many examples comming from statistical mechanics, we construct a formal asymptotic expansion for Λ(h) in powers of h, from which we derive precise estimates. We work in the setting of standard functions introduced by J. Sjöstrand for the study of similar questions in the case of Schrödinger operators.

The following sections are included:

• INTRODUCTION

• METHOD OF THE PROOF

• EXPRESSION OF BROKEN RAYS CONVERGING TO THE PERIODIC RAY

• SOLUTIONS OF FUNCTION EQUATIONS

• TAYLOR EXPANSION OF Tⁿ(s + t, σ + τ)

• REFERENCES

The following sections are included:

• Preliminaires

• Parabolic rational maps
  • ANALYTIC PROPERTIES OF THE PRESSURE
  • ESCAPE RATE
  • ASYMPTOTIC DISTRIBUTION OF PREIMAGES

• References

We introduce a class of noncompact Riemannian manifolds on which we can argue quantum field theory for scalar particles in external fields. More precisely, we consider quantized linear Klein-Gordon fields subject to (non quantized) electro-magnetic forces in a certain class of static space-time. This class is broad enough to include physically important examples of the Euclidean space, the hyperbolic space, and by passing to the natural Lorentzian structure, the Schwarzschild metric up to conformal equivalence. The S-matrix of the massive Klein-Gordon equation on these manifolds is unitarily implemented on the Fock space constructed via the spectrum of the Laplace-Beltrami operator with scalar curvature. We also give the same result for the massless case in the asymptotically flat and hyperbolic spaces.

We prove the finiteness of the total scattering cross-section for ion-atom collisions with an initial channel given by a simple eigenvalue of the internal Hamiltonian describing the neutral cluster, i.e. the atom. Under more restrictive assumptions, we show that some effective interaction in Born-Oppeheimer approximation is precisely of order in the distance between the mass centers of two clusters. We then extract the leading term of the scattering cross-section in the Born-Oppenheimer limit.

Results concerning singular perturbation theory for Quantum Mechanics are presented. The framework is a generalisation of the spectral concentration theory. Under very general conditions, asymptotic estimations on the Rayleigh-Schrödinger expansions of the perturbed spectral projections are obtained. As a consequence almost invariant subspaces of exponential order are constructed. In the magnetic field case, under the condition that the magnetic field does not increase at infinity, a gauge invariant perturbation theory leading to convergent series with field dependent coefficients is developed.

The following sections are included:

• Introduction

• Assumptions

• Main result

• The explicit model

• Splitting vanishing and localization

• Bibliography

In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kähler manifolds are presented. These results are obtained in joint work with M. Bordemann and E. Meinrenken. The existence of the Berezin-Toeplitz deformation quantization is also covered. Recent results obtained in joint work with A. Karabegov on the asymptotic expansion of the Berezin transform for arbitrary quantizable compact Kähler manifolds are explained. As an application the asymptotic expansion of the Fubini-Study fundamental form under the coherent state embedding is considered. Some comments on the dynamics of the quantum operators are given.