Program: (the order may change)
8h45: Welcoming
9h00-9h50: Shimon Brooks (Institute for Mathematical Sciences, Stony Brook University) web page
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) web page
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: Coffee break
11h30-12h20: Luc Hillairet (Laboratoire Jean Leray, Nantes) web page
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: Lunch break
14h30-15h20: Rémy Dubertrand (Institut für Theoretische Physik, Heidelberg) web page
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: Coffee break
15h50-16h40: Brian Winn (School of Mathematics, Loughborough University) web page
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.
