Participants
Programme: (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
14h30-15h20: 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.
