Spectral and scattering theory of massive Pauli-Fierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of observables. Its main result is what we call geometric asymptotic completeness. Finally, the last part is a proof of asymptotic completeness itself. 1 Introduction Our paper is devoted to a class of Hamiltonians used in physics to describe a quantum system ("matter" or "an atom") interacting with a bosonic field ("radiation"). K and K are respectively the Hilbert space and the Hamiltonian describing the matter. The bosonic field is described by a Fock space \Gamma(h) with the one-particle space eg. h = L 2 (IR d ; dk), where IR d is the momentum space, and a free Hamiltonian of the form d\Gamma(!(k)) = Z !(k)a (k)a(k)dk: The function !(k) is called the dispersion relation. The inte...

The purpose of this article is to discuss a simple linear algebraic tool which has proved itself very useful in the mathematical study of spectral problems arising in elecromagnetism and quantum mechanics. Roughly speaking it amounts to replacing an operator of interest by a suitably chosen invertible system of operators.

Let P = −h 2 ∆g +V (x) be a self-adjoint Schrödinger operator on a compact Riemannian n-manifold, (X, g), V ∈ C ∞ (X; R). The spectral asymptotics as h → 0 are given by the celebrated Weyl law – see [10] and [16] for recent advances and numerous references. If we assume that the zero energy surface is nondegenerate,

One of the interesting developments in spectral theory recently has been the application of a new formula for a function j{H) of a self-adjoint operator H by Helffer and Sjostrand [8]. This formula, equation (5) below, depends upon the use of almost analytic extensions of functions initially defined on the real line, an idea due

Abstract. In this paper we describe the propagation of singularities of tempered distributional solutions u ∈ S ′ of (H − λ)u = 0, λ> 0, where H is a many-body Hamiltonian H = ∆ + V, ∆ ≥ 0, V = P a Va, under the assumption that no subsystem has a bound state and that the two-body interactions Va are real-valued polyhomogeneous symbols of order −1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term ‘singularity ’ provides a microlocal description of the lack of decay at infinity. We use this result to prove that the wave front relation of the free-to-free S-matrix (which, under our assumptions, is all of the S-matrix) is given by the broken geodesic flow, broken at the ‘singular directions’, on Sn−1 at time π. We also present a natural geometric generalization to asymptotically Euclidean spaces. Propagation des singularités pour la diffusion dans le problème à N corps Résumé. Dans cet article on décrit la propagation des singularités des solutions tempérées u ∈ S ′ de (H − λ)u = 0, λ> 0, où H est un Hamiltonien à N corps H = ∆ + V, ∆ ≥ 0, V = P a Va, en supposant que les Hamiltoniens des sous-systèmes n’ont pas de vecteurs propres (dans L2), et que les potentiels à deux corps Va sont des symboles polyhomogènes réels d’ordre −1 (par exemple, de type Coulomb, mais sans la singularité à l’origine). Ici le terme “singularité” fournit une description microlocale de la croissance des fonctions à l’infini. On emploie ce résultat pour montrer que la relation de front d’onde de la matrice de diffusion, N-amas N-amas (qui est la seule partie de la matrice de diffusion sous nos hypothèses), est donnée par le flot géodesique brisé dans les “directions singulières”, sur Sn−1 à temps π. On présente aussi une généralisation géometrique naturelle au cas des variétés asymptotiquement euclidiennes. 1.

. We give a full and selfcontained account of the basic results in N--body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r \Gamma¯ , ¯ ? p 3 \Gamma 1 . Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t 2 as t ! \Sigma1 for any orbit / t which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase--space analysis. As a topic of general interest we describe a method of commutator expansions. 0. INTRODUCTION N-body quantum systems are described by the Schrodinger equation i@ t / t = H/ t , with a Hamiltonian like...

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of Mathematical Physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

Abstract. We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the non-interacting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the well-known Kubo-St˘reda formula for the quantum Hall conductivity at zero temperature.

In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle ” at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.

We analyze in this article the spectral properties of the Schrodinger operator with periodic potential on L 2 (IR n ). It is proven that the integrated density of states N(¯) has an asymptotic expansion of the form N(¯) = a n ¯ n=2 + a n\Gamma2 ¯ n\Gamma2 2 +O(¯ (n\Gamma3+ffl)=2 ); 8ffl ? 0 : This gives also a proof of the Bethe-Sommerfeld conjecture for n 4. 1 Introduction We are interested in the study of the spectral properties of the Schrodinger operator on L 2 (IR n ); with n 2; P V = n X j=1 D 2 x j + V (x); (1.1) where D x j = \Gammai@ x j = 1 i @ @x j , for j = 1; \Delta \Delta \Delta ; n. The potential x 7! V (x) is real and V (\Delta) 2 C 1 (IR n ; IR): (1.2) Let \Gamma be a lattice on IR n : \Gamma = f n X j=1 k j e j ; k = (k 1 ; : : : ; k n ) 2 ZZ n g ; (1.3) where fe 1 ; : : : ; e n g is a basis of IR n . We assume that the potential is periodic: V (x + a) = V (x) ; 8a 2 \Gamma : (1.4) It is well known that P V is essentiall...