Asymptotic Analysis - Volume 40, issue 2

Authors: Adami, Riccardo | Bardos, Claude | Golse, François | Teta, Alessandro

Article Type: Research Article

Abstract: We consider a system of N particles in dimension one, interacting through a zero‐range repulsive potential whose strength is proportional to N−1 . We construct the finite and the infinite Schrödinger hierarchies for the reduced density matrices of subsystems with n particles. We show that the solution of the finite hierarchy converges in a suitable sense to a solution of the infinite one. Besides, the infinite hierarchy is solved by a factorized state, built as a tensor product of many identical one‐particle wave functions which fulfil the cubic nonlinear Schrödinger equation. Therefore, choosing a factorized initial datum and assuming propagation …of chaos, we provide a derivation for the cubic NLSE. The result, achieved with operator‐analysis techniques, can be considered as a first step towards a rigorous deduction of the Gross–Pitaevskii equation in dimension one. The problem of proving propagation of chaos is left untouched. Show more

Citation: Asymptotic Analysis, vol. 40, no. 2, pp. 93-108, 2004

Authors: Colli, Pierluigi | Shirakawa, Ken

Article Type: Research Article

Abstract: In this paper, we consider a one‐dimensional Frémond model of shape memory alloys. Let us imagine a wire of a shape memory alloy whose left‐hand side is fixed, and assume that forcing terms, e.g., heat sources and external stress on the right‐hand side, converge to some time‐independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we first show the existence of the global attractor for the limiting autonomous dynamical system, for instance the case of zero …external stress, and secondly, characterize the asymptotic stability for nonautonomous case by the limiting global attractor. Show more

Citation: Asymptotic Analysis, vol. 40, no. 2, pp. 109-135, 2004

Authors: Dimassi, Mouez | Guillot, Jean‐Claude | Ralston, James

Article Type: Research Article

Abstract: We construct almost invariant subspaces and the corresponding effective Hamiltonian for magnetic Bloch bands. We also discuss the question of the dynamics related to the effective Hamiltonian. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively.

Citation: Asymptotic Analysis, vol. 40, no. 2, pp. 137-146, 2004

Authors: Boulkhemair, A.

Article Type: Research Article

Abstract: L2 continuity results for Fourier integral operators with nonregular phases and amplitudes are established.

Citation: Asymptotic Analysis, vol. 40, no. 2, pp. 147-161, 2004

Authors: Benedetto, D. | Esposito, R. | Pulvirenti, M.

Article Type: Research Article

Abstract: We consider a quantum particle scattered by an obstacle localized around the origin. We look at the time evolution in a large space‐time scale, O(ε−1 ), assuming as initial datum a suitable WKB state with an amplitude varying on scale ε−1 . We study asymptotic behavior of the solution when ε vanishes. Our analysis is complementary to the one performed by Nier (Transport Theory Statist. Phys. 24(4–5) (1995), 591–629; Ann. Sci. École Norm. Sup. (4) 29 (1996), 149–183) who studied the same problem directly in terms of the Wigner formalism.

Citation: Asymptotic Analysis, vol. 40, no. 2, pp. 163-187, 2004