分野別セミナー

アブストラクト：
We will present in a parallel way, some questions and results in "
classical chaos" and "quantum chaos" in the simplest case of hyperbolic
dynamics. The main questions that will concern us are long time behavior
and the spectrum of stationnary states. We will present some standard
models such as dynamics of billards, geodesic flow on a surface with
negative curvature, or hyperbolic diffeomorphisms on a compact manifold.

In the case of geodesic flow for example, in "classical chaos" we are
interested in the spectrum of the vector field which generates the flow
(so called Ruelle resonances), and which governs the long time
convergence towards equilibrium. Notice that this discrete spectrum
provides a simple paradigm for emerging irreversibility in mechanics
when fundamentals microscopic laws are deterministic. The related model
in "quantum chaos" is the wave equation and the spectrum of the
Laplacian on this surface. The main results in quantum chaos are the "
Weyl formula" for the smooth density of states, the "quantum ergodicity
or Schnirelman theorem" concerning the equidistribution of stationnary
states , the "Gutzwiller trace formula" which relates periodic classical
orbits to the density of states, and the "Random matrix theory" (this
later is still conjectural). We will explain how these results are
related to different time regime and how they could be transposed in "
classical chaos".
In the second part of the talk we will present in more details, a recent
semiclassical approach to the Ruelle resonances spectrum, a joined work
with Nicolas Roy and Johannes Sjöstrand. The aim of this work is to
make closer the relations between "classical chaos" and "quantum chaos".