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Documents  35Q55 | enregistrements trouvés : 10

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Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.

I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I will argue that, under certain circumstances, these become anyons. This is made manifest by the emergence of a particular effective Hamiltonian for their motion. The latter is notoriously hard to solve even in simple cases, and well-controled simplifications are highly desirable. I will discuss a possible mean-field approximation, leading to a one-particle energy functional with self-consistent magnetic field.
Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.

I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I ...

82B10 ; 81S05 ; 35P15 ; 35Q40 ; 35Q55 ; 81V70

We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite temperature. We will also describe the numerical methods which have been developed for those models in the framework of the ANR project Becasim. These are joint works with Reika Fukuizumi, Arnaud Debussche, and Romain Poncet. We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite ...

35Q55 ; 60H15 ; 65M06

quasi-periodic wave equation - laplacian - hamiltonian system - symplectic form - time dependance

37K55 ; 35Q55

To illustrate specifically quantum behaviours, the talk will consider three typical problems for non-linear kinetic models evolving through pair collisions at temperatures not far from absolute zero. Based on those examples, a number of differences between quantum and classical Boltzmann theory is discussed in more general term.

82D50 ; 76Y05 ; 82D30 ; 35Q60 ; 35Q55

We will discuss the convergence (in the semiclassical limit) of a solution to the Hartree-Fock equation towards an operator, whose Wigner transform is a solution to the Vlasov equation. We will consider both cases of positive and zero temperature. The results we will present are part of a project in collaboration with N. Benedikter, M. Porta and B. Schlein.

82C22 ; 82C10 ; 35Q40 ; 35Q55

We consider the nonlinear Schrödinger equation in the partially periodic setting $\mathbb{R}^d\times \mathbb{T}$. We present some recent results obtained in collaboration with N. Tzvetkov concerning the Cauchy theory and the long-time behavior of the solutions.

nonlinear Schrödinger equation - Cauchy theory - scattering

35Q55 ; 35B40 ; 35P25

This talk is devoted to the study of the following inverse boundary value problem: given a Riemannian manifold with boundary determine the magnetic potential in a dynamical Schrödinger equation in a magnetic field from the observations made at the boundary.

inverse problem - Schrödinger equation - magnetic field

35R30 ; 35Q55 ; 35R35 ; 35Q60

We consider solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss different extensions of this result to equations with bounded potentials. The continuous case was studied in a series of articles by L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega.
The talk is mainly based on joint work with Ph. Jaming, Yu. Lyubarskii, and K.-M. Perfekt.
We consider solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss ...

35Q55 ; 35A02

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation. Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzb...

35Q55 ; 35Q56 ; 82D55

The leapfrogging is the name given to a regime of interaction between vortex rings with the same axis of symmetry in incompressible fluids. We will explain where it comes from and indicate a rigorous derivation in the case of the axisymmetric Gross-Pitaevskii equation.

35Q55 ; 35Q56 ; 76B47

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