We develop the L p -theory of the spectral shift function, for p ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace ideal I p , for 0 < p ≤ 1. This result is a key ingredient of a new, short proof of the Wegner estimate applicable to a wide variety of additive and multiplicative random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. Under an additional condition of the single-site potential, local Hölder continuity is proved at all energies. This new Wegner estimate, together with other, standard results, establishes exponential localization for a new family of models for additive and multiplicative perturbations.

Abstract. We improve the Lieb-Robinson bound for a wide class of quantum many-body systems with long-range interactions decaying by power law. As an application, we show that the group velocity of information propagation grows by power law in time for such systems, whereas systems with short-range interactions exhibit a finite group velocity as shown by Lieb and Robinson.

In the present note, we determine the ground state energy and study the existence of Lifshitz tails near this energy for some non monotonous alloy type models. Here, non monotonous means that the single site potential coming into the alloy random potential changes sign. In particular, the random operator is not a monotonous function of the random variables.Résumé. Cet article est consacréà la détermination de l'énergie de l'état fondamental età l'étude de possibles asymptotiques de Lifshitz au voisinage de cetteénergie pour certains modèles d'Anderson continus non monotones. Ici, non monotone signifie que le potentiel de simple site entrant dans la composition du potentiel aléatoire change de signe. En particulier, l'opérateur aléatoire n'est pas une fonction monotone des variables aléatoires. Introduction and resultsIn this paper, we consider the continuous alloy type (or Anderson) random Schrödinger operator:where V is the site potential, and (ω γ ) γ∈Z d are the random coupling constants. Throughout this paper, we assume Let Σ be the almost sure spectrum of H ω and E − = inf Σ. When V has a fixed sign, it is well known that theMoreover, in this case, it is well known that the integrated density of states of the Hamiltonian (see e.g. (0.3)) admits a Lifshitz tail near E − , i.e., that the integrated density of states at energy E decays exponentially fast as E goes to E − from above. We refer to [9,7,22,20,6,5,11] for precise statements.In the present paper, we address the case when V changes sign, i.e., there may exist x + = x − such thatThe basic difficulty this property introduces is that the variations of the potential V ω as a function of ω are not monotonous. In the monotonous case, to get the minimum, one can simply minimize with respect to each of the random variables individually. In the non monotonous case, this uncoupling between the different random variables may fail. Our results concern reflection symmetric potentials since, as we will see, for these potentials we also have an analogous decoupling between the different random variables. Thus, we make the following symmetry assumption on V :We now consider the operator H N λ = −∆ + λV with Neumann boundary conditions on the cube [−1/2, 1/2] d . Its spectrum is discrete, and we let E − (λ) be its ground state energy. It is a simple eigenvalue and λ → E − (λ) is a real analytic concave function defined on R. We first observe:Proposition 0.1. Under the above assumptions (H1) and (H2),For a and b sufficiently small, this result was proven in [17] without the assumption (H2) but with an additional assumption on the sign ofThe method used by Najar relies on a small coupling constant expansion for the infimum of Σ. These ideas were first used in [3] to treat other non 2 monotonous perturbations, in this case magnetic ones, of the Laplace operator. In [1], the authors study the minimum of the almost sure spectrum for a random displacement model i.e. the random potential is defined as V ω (x) = γ∈Z d V (x − γ − ξ γ ) where (ξ γ ) γ are i.i.d. rando...

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Sufficient conditions are given for the boundedness of f(H), H = -!::::. + V, in £P(Rd), 1 :' .S p ' .S oo. Optimal results with respect to the decay off are obtained for LP-boundedness of e-itH f(H) and the nearly-optimal norm-estimate lle-itH f(H)lls(LP) :' .S C(l + ltl)',Results are also obtained on the mapping properties of e-itH between certain Besov spaces.

In this paper we study the propagation of singularity for Schrödingertype equations with variable coefficients. We introduce a new notion of wave propagation set, the homogeneous wave front set, and it propagates along straight lines with finite speed away from x = 0. Then we show that it is related to the wave front set in a natural way. These results may be considered as a refinement of the microlocal smoothing property of Craig-Kappeler-Strauss under more general assumptions.

We study Lifshitz tails for random Schrödinger operators where the random potential is alloy-type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose the single site potentials are distributed in a finite set of functions, and we show that under suitable symmetry conditions, they have a Lifshitz tail at the bottom of the spectrum except for special cases. When the single site potential is symmetric with respect to all the axes, we give a necessary and sufficient condition for the existence of Lifshitz tails. As an application, we show that certain random displacement models have a Lifshitz singularity at the bottom of the spectrum, and also complete our previous study (2009) of continuous Anderson type models.

We consider Schrödinger equations with variable coefficients, and it is supposed to be a long-range type perturbation of the flat Laplacian on R n . We characterize the wave front set of solutions to Schrödinger equations in terms of the initial state. Then it is shown that the singularity propagates following the classical flow, and it is formulated in a semiclassical setting. Methods analogous to the long-range scattering theory, in particular a modified free propagator, are employed.