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# Documents  35Q55 | enregistrements trouvés : 10

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## Post-edited  Emergent anyons in quantum Hall physics Rougerie, Nicolas (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## Post-edited  Inhomogeneities and temperature effects in Bose-Einstein condensates de Bouard, Anne (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## Multi angle  Quasi-periodic wave equation - almost reducibility - Eliasson, Hakan (Auteur de la Conférence) | CIRM (Editeur )

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

## Multi angle  On low temperature kinetic theory; spin diffusion, anyons, Bose Einstein condensates Arkeryd, Leif (Auteur de la Conférence) | CIRM (Editeur )

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.

## Multi angle  From the Hartree-Fock dynamics to the Vlasov equation Saffirio, Chiara (Auteur de la Conférence) | CIRM (Editeur )

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.

## Multi angle  Scattering for NLS in $\mathbb{R}^d\times \mathbb{T}$ Visciglia, Nicola (Auteur de la Conférence) | CIRM (Editeur )

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

## Multi angle  Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field Bellassoued, Mourad (Auteur de la Conférence) | CIRM (Editeur )

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

## Multi angle  Uncertainty principles for discrete Schrödinger evolutions Malinnikova, Eugenia (Auteur de la Conférence) | CIRM (Editeur )

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 ...

## Multi angle  Mean field limits for Ginzburg-Landau vortices Serfaty, Sylvia (Auteur de la Conférence) | CIRM (Editeur )

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...

## Multi angle  Leapfrogging for the axisymmetric Gross-Pitaevskii equation Smets, Didier (Auteur de la Conférence) | CIRM (Editeur )

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.

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