We develop a polymer expansion with large/small field conditions for the mean resolvent of a weakly disordered system. Then we show that we can apply our result to a two-dimensional model, for energies outside the unperturbed spectrum or in the free spectrum provided the potential has an infra-red cut-off. This leads to an asymptotic expansion for the density of states. We believe this is an important first step towards a rigourous analysis of the density of states in the free spectrum of a random Schrödinger operator at weak disorder.
Symmetries are automorphisms of the observable algebra A. There are two notions of spontaneous symmetry breaking. The stronger refers to a state ω over A such that the symmetry does not extend to the strong closure πω(A)″ where πω is the GNS-representation associated to ω. The standard example for that are the Schwinger terms [j(x), j(x′)]=iδ′(x−x′), x ∈ R which are not invariant under the parity j(x) → j(−x) though the j’s are constructed from fermion fields ψ(x) as strong limits and ψ(x) → ψ(−x) is a symmetry of the fermion algebra. The weaker notion which can also be realized in elementary quantum mechanics refers to a time evolution τt ∈ Aust A. A symmetry σ is said to be spontaneously broken if there is a state ω with ω o τt=ω, ω o σ≠ω though [σ, τt]=0.
We discuss relations between different formulae for solutions of the Knizhnik-Zamolodchikov differential and the quantum Knizhnik-Zamolodchikov difference equations at level 0 and associated with rational solutions of the Yang-Baxter equation.
We introduce spatial disorder in a large system of interacting particles that evolve according to a non-reversible dynamical law. We show that if the regions where the components strongly interact are scarce, several general properties of the discrete and continuous time dynamics remain unaffected by the disorder.For the discrete time dynamics we prove that the unique invariant measure is Gibbsian, its two-point spatial correlation function decays exponentially fast for increasing distances and, for a restricted class of models (i.e., directed probabilistic cellular automata), we prove almost sure and disorder-averaged upper bounds for the rate of relaxation towards equilibrium. Moreover we show, by an example, that under our conditions these bounds are (almost) optimal.For the continuous time dynamics, after showing the existence of the infinite volume limit, we derive approximations by a discrete time updating system, valid uniformly in time. (C) Elsevier, Paris.
Nous etudions la propagation des ondes acoustiques et electromagnetiques dans les milieux aleatoires. Le modele est decrit par un operateur non-perturbe avec un gap dans le spectre. Ajoutant une perturbation aleatoire, nous montrons l'existence des ondes localisees aux energies aux bords du spectre de l'operateur non-perturbe presque surement. En plus, nous donnons une demonstration de l'existence et la continuite lipschitzienne de la densite d'etats integree, suivant une nouvelle estimation de type Wegner.
We consider a perturbed Floquet Hamiltonian $-i\partial_t + H + \beta V(\omega t)$ in the Hilbert space $L^2([0,T],E,dt)$. Here $H$ is a self-adjoint operator in $E$ with a discrete spectrum obeying a growing gap condition, $V(t)$ is a symmetric bounded operator in $E$ depending on $t$ $2\pi$-periodically, $\omega = 2\pi/T$ is a frequency and $\beta$ is a coupling constant. The spectrum $Spec(-i\partial_t + H)$ of the unperturbed part is pure point and dense in $R$ for almost every $\omega$. This fact excludes application of the regular perturbation theory. Nevertheless we show, for almost all $\omega$ and provided $V(t)$ is sufficiently smooth, that the perturbation theory still makes sense, however, with two modifications. First, the coupling constant is restricted to a set $I$ which need not be an interval but 0 is still a point of density of $I$. Second, the Rayleigh-Schrodinger series are asymptotic to the perturbed eigen-value and the perturbed eigen-vector.
We consider the complete normal field net with compact symmetry group constructed by Doplicher and Roberts starting from a net of local observables in >=2+1 spacetime dimensions and its set of localized (DHR) representations. We prove that the field net does not possess nontrivial DHR sectors, provided the observables have only finitely many sectors. Whereas the superselection structure in 1+1 dimensions typically does not arise from a group, the DR construction is applicable to `degenerate sectors', the existence of which (in the rational case) is equivalent to non-invertibility of Verlinde's S-matrix. We prove Rehren's conjecture that the enlarged theory is non-degenerate, which implies that every degenerate theory is an `orbifold' theory. Thus, the symmetry of a generic model `factorizes' into a group part and a pure quantum part which still must be clarified.
We prove that if the non-self-adjoint scalar wave equation satisfies Huygens' principle on Petrov type III space-times, then it is equivalent to the conformally invariant scalar wave equation.
We prove that there are no Petrov type III space-times on which the conformally invariant (self-adjoint) scalar wave equation or the non-self-adjoint scalar wave equation satisfies Huygens' principle.
Fluctuations of observables as functions of time, or "fluctuation patterns", are studied in a chaotic microscopically reversible system that has irreversibly reached a nonequilibrium stationary state. Supposing that during a certain, long enough, time interval the average entropy creation rate has a value $s$ and that during another time interval of the same length it has value $-s$ then we show that the relative probabilities of fluctuation patterns in the first time interval are the same as those of the reversed patterns in the second time interval. The system is ``conditionally reversible'' or irreversibility in a reversible system is "driven" by the entropy creation: while a very rare fluctuation happens to change the sign of the entropy creation rate it also happens that the time reversed fluctuations of all other observables acquire the same relative probability of the corresponding fluctuations in presence of normal entropy creation. A mathematical proof is sketched.
Partant de l'hypothese que les etats de vide dans un espace de De Sitter sont percus par tout observateur geodesique comme des etats d'equilibre ayant a priori une temperature quelconque, nous analysons leurs proprietes globales dans le cadre algebrique de la physique quantique locale. Nous montrons que ces etats possedent la propriete de Reeh-Schlieder et que tout etat de vide primitif est pur et faiblement melangeant. De plus, la temperature geodesique d'un etat de vide doit obligatoirement coincider avec celle de Gibbons-Hawking, propriete reliee etroitement a l'exterieur d'une symetrie discrete du type PCT. Nous montrons enfin que les algebres d'observables globales dans les secteurs de vide ont la meme structure que leurs analogues dans les theories d'espace de Minkowski.
We consider the Schrodinger operator P-V(h) = -h(2)Delta + V where V is an element of C-0(R-n) such that lim(\\x\\-->+infinity) V(x) = +infinity. For every phi convex with a support in R+, we state the following inequality (*) Tr(phi(E - P-V(h))) less than or equal to h(-n)/(2 pi)(n) integral(Rn)integral(Rn) phi(E - xi(2) - V(x)) dx d xi for all E is an element of R and all h is an element of R+, when V is strictly convex and quadratic. When phi = max{t, 0}(gamma) gamma greater than or equal to 1 and n greater than or equal to 3, the inequality (*) is the Lieb-Thirring's conjecture, (C) Elsevier, Paris.
A small enough magnetic field on a compact riemannian manifold with negative curvature generates an Anosov perturbation of the geodesic flow. The topological entropy ratio is explicitly hounded. The entropy of the Liouville measure admits an Osserman-Sarnak estimate. In dimension two, for a magnetic field which is small and regular enough, and has zero mean, the flow can be parametrized by the action of the lagrangian. The topological entropy is shown to be greater than the Liouville metric entropy, unless the magnetic field is zero and the Gauss curvature is constant; this extends a theorem of Katok known for the geodesic flow. (C) Elsevier, Paris.
In relation with the Born-Oppenheimer approximation, we study the scattering operator S associated to a 2 x 2 semiclassical matricial Schrodinger operator, near a non-trapping energy level. Under some gap condition and assumptions of analyticity and decay at infinity, we show that the two off-diagonal elements of S are exponentially small as the semiclassical parameter tends to zero. Moreover, the rate of exponential decay can be explicited depending on the behaviour in the complex domain of the two electronic levels. (C) Elsevier, Paris.
We briefly review the main aspects of (Poincaré–Dulac) normal forms; we have a look at the nonuniqueness problem, and discuss one of the proposed ways to ‘further reduce’ the normal forms. We also mention some convergence results.
The aim of this study is to estimate the residu of the scattering matrix associated with resonances for two-body Schrodinger operators with long range interactions in the semiclassical approach. We assume that there are no caustics out of the classical forbidden area surrounding the well. Then we obtain a semiclassical BKW expansion of the resonating states in a neighborhood of some vanishing bicharacteristics of the operator. This expansion allows us to find out an estimate of the residu of the kernel of the scattering matrix, which appears to be either of the same order as the width of the resonance, or such depending on the fact that (omega, omega') belongs or not to a certain set explicitely described. (C) Elsevier, Paris.
In the first part of this work, we compute explicitly the principal symbol of a WKB solution of the Dirac operator along a planar Agmon geodesic starting from a non-degenerate ponctual well. In the second part, we apply this result to study the band spectrum of a periodic Dirac operator near the minimum of the potential, we construct a class of potentials where the multiplicity of the Floquet eigenvalues is 1 and not 2 (eigenvalues of a Dirac operator with zero magnetic field have even multiplicity). (C) Elsevier, Paris.
New representations of the Poincare group are given, which describe two bosons with interaction in four space-time dimensions. The quantum frame is the Schrodinger picture in momentum space. More precisely we start from the relativistic free model with Hilbert space L-2(R-3 x R-3, sigma(2)), where sigma(2) is the Lorentz invariant measure. We add to the free Hamiltonian and the free Lorentz generators new interaction terms, without changing the Poincare algebra commutation rules, and such that the algebra representation can be integrated to give a unitary representation of the group on L-2(R-6, sigma(2)) The physics of these models can be investigated through the bound state equation (a relativistic Schrodinger equation) and through the scattering matrix, which is shown to be unitary. Finally we give an example for which a bound state exists and for which the scattering matrix can be written down explicitely. This example assures that interaction can effectively occur between the particles. (C) Elsevier, Paris.
Le but de cette etude est l'estimation semiclassique des residus de la matrice de diffusion S(λ) associes a des resonances de forme, pour des operateurs de Schrodinger a deux corps P = -h 2 Δ + V(x) ou V est un potentiel a longue portee. En l'absence de caustiques, nous obtenons un developpement B.K.W a l'infini de l'etat resonnant au voisinage de toute bicaracteristique de P. Grâce a ce developpement, nous obtenons une estimation du residu du noyau S(λ, h, ω, ω') de S(λ), qui se trouve etre soit comparable a la largeur de la resonance, soit beaucoup plus petit selon que (ω, ω') appartient ou non a un certain ensemble explicitement decrit.
En relation avec l'approximation de Born-Oppenheimer, on etudie l'operateur de diffusion S associe a un operateur de Schrodinger matriciel 2 x 2, pres d'un niveau d'energie non-captif. Sous une condition de gap et des hypotheses d'analyticite et de decroissance a l'infini, on montre que les deux elements non-diagonaux de S deviennent exponentiellement petits lorsque le parametre semiclassique tend vers zero. De plus, le taux de decroissance peut etre explicite en fonction du comportement dans le complexe des deux niveaux electroniques.
