Une journée autour du chaos quantique
15 décembre 2010
Laboratoire Paul Painlevé (Université de Lille 1)

English version

Organisateurs: Stephan De Bièvre (Laboratoire Paul Painlevé, Université Lille 1), Stéphane Nonnenmacher (IPHT, CEA Saclay), Gabriel Rivière (Laboratoire Paul Painlevé, Université Lille 1)

Inscriptions: Merci de contacter Gabriel Rivière pour vous inscrire (gabriel.riviere(le symbole)math.univ-lille1.fr)


(l'ordre peut changer)

8h45: Accueil

9h00-9h50: Shimon Brooks (Institute for Mathematical Sciences, Stony Brook University) page web
                Spectral multiplicities and Quantum Unique Ergodicity.
The Quantum Unique Ergodicity (QUE) property--- that all eigenstates become equidistributed in the semiclassical limit--- is conjectured to hold for the Schrödinger evolution on manifolds of negative sectional curvature, but is known to fail for some toy models of quantum chaos; eg., for quantized cat maps. The latter are known to exhibit large spectral degeneracies, and it is thought that this could be causing QUE to fail. For Riemann surfaces, we will give a precise conjecture on the multiplicity bounds required for QUE, and discuss some evidence in this direction, in light of recent joint work with E. Lindenstrauss on the arithmetic case. We will also discuss a similar conjecture for cat maps.

10h00-10h50: Frédéric Faure (Institut Fourier, Grenoble) page web
                Upper bound on the density of Ruelle resonances for Anosov flows.
Uniformly hyperbolic flow (also called Anosov flow) on a compact manifold is a standard model of "chaotic classical dynamics". In this talk we will present a common work with Johannes Sjöstrand (arXiv:1003.0513v1). Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real axis and for large real parts. One objective of this work is to make more precise the connection between the spectral study of Ruelle resonances and the spectral study in quantum chaos.

10h50-11h30: Pause café

11h30-12h20: Luc Hillairet (Laboratoire Jean Leray, Nantes) page web
                Adiabatic approximation and semiclassical concentration.
We present several methods to address (non-)concentration of eigenfunctions in stadium-like billiards. One of which corresponds to an approximate separation of variables known as adiabatic approximation.

12h20-14h30: Pause déjeuner

Rémy Dubertrand (Institut für Theoretische Physik, Heidelberg) page web
                Semiclassical technics for dielectric cavities.
We present some numerical results for dielectric cavities. They are a kind of open billiards, which can be realized in experiments. The focus will be taken on a statistical analysis of the spectrum for simple shapes. Some functions attached to these resonances will also be shown.

15h20-15h50: Pause café

15h50-16h40: Brian Winn (School of Mathematics, Loughborough University) page web
                Localised eigenfunctions in Šeba billiards.
We describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application we find, under some additional hypotheses on the spectrum, subsequences of eigenfunctions of Šeba billiards that localise around a pair of unperturbed eigenfunctions.

Renseignements pratiques:
Soutiens Financiers:
G.D.R. Dynamique Quantique, Laboratoire Paul Painlevé, ANR Methchaos, CNRS, Université de Lille 1