The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the $x_{3}$-axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the $x_{3}$-axis, we consider the reduced Hamiltonian associated with the total momentum along the $x_{3}$-axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the $x_{3}$-axis are sufficiently small. Finally we determine the absolutely continuous spectrum of the reduced Hamiltonian.Comment: typos correction

Inverse scattering at fixed energy for layered media

AbstractIn this article we show that exponentially decreasing perturbations of the sound speed in a layered medium can be recovered from the scattering amplitude at fixed energy. We consider the unperturbed equation utt = c02(xn)δu in ℝ×ℝ, where n ≥ 3. The unperturbed sound speed, c0(xn), is assumed to be bounded, strictly positive, and constant outside a bounded interval on the real axis. The perturbed sound speed, c(x), satisfies ¦c.(x) - co(xn)¦ < C exp(−δ¦x¦) for some δ > 0. Our work is related to the recent results of H. Isozaki (J. Diff. Eq. 138) on the case where c0 takes the constant values c+ and c− on the positive and negative half-lines, and R. Weder on the case c0 = c+ for xn > h, c0 = ch, for 0 < xn, < h, and c0 = c− for xn < 0 (IIMAS-UNAM Preprint 70, November, 1997)

Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-. I

We consider a Hamiltonian with cutoffs describing the weak decay of spin one massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval.Comment: Correction of minor misprint

Effect of turbulence on collisions of dust particles with planetesimals in protoplanetary disks

Planetesimals in gaseous protoplanetary disks may grow by collecting dust particles. Hydrodynamical studies show that small particles generally avoid collisions with the planetesimals because they are entrained by the flow around them. This occurs when $St$, the Stokes number, defined as the ratio of the dust stopping time to the planetesimal crossing time, becomes much smaller than unity. However, these studies have been limited to the laminar case, whereas these disks are believed to be turbulent. We want to estimate the influence of gas turbulence on the dust-planetesimal collision rate and on the impact speeds. We used three-dimensional direct numerical simulations of a fixed sphere (planetesimal) facing a laminar and turbulent flow seeded with small inertial particles (dust) subject to a Stokes drag. A no-slip boundary condition on the planetesimal surface is modeled via a penalty method. We find that turbulence can significantly increase the collision rate of dust particles with planetesimals. For a high turbulence case (when the amplitude of turbulent fluctuations is similar to the headwind velocity), we find that the collision probability remains equal to the geometrical rate or even higher for $St\geq 0.1$, i.e., for dust sizes an order of magnitude smaller than in the laminar case. We derive expressions to calculate impact probabilities as a function of dust and planetesimal size and turbulent intensity

Coalescence in the 1D Cahn-Hilliard model

We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing continuously the processus of destabilization and period doubling predicted in Langer's self similar scenario

Twisting algebras using non-commutative torsors

Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori.Comment: 27 pages. Masuoka is a new coauthor. Introduction was revised. Sections 1 and 2 were thoroughly restructured. The presentation theorem in Section 3 is now put in a more general framework and has a more general formulation. Section 4 was shortened. All examples (quantum affine spaces and tori, twisting of SL(2), twisting of the enveloping algebra of sl(2)) are left unchange

Quelques commentaires sur la linéarisation de l'erreur de l'observation multi-sorties

International audienceCe papier donne quelques idées et commentaires sur les conditions géométriques qui permettent de dire si un système non linéaire multi sorties possède, à un changement de coordonnées près, une erreur d'observation linéaire. Plus précisément, nous allons concentrer notre point de vu sur les travaux de Krener et Respondek d'une part et ceux de Xia et Gao d'autre part. Nous allons aussi commenter le "soit disant" contre exemple de Xia et Gao. Puis, nous allons présenter l'algorithme de la linéarisation de l'erreur de l'observation par extension dynamique