Tel-Aviv University
School of Mathematical Sciences

Department Colloquium

Monday, January 7, 2013

Schreiber 006, 12:15

Stéphane Nonnenmacher 

Institut de Physique Théorique, CEA-Saclay

TIDY Distinguished lecture in Mathematical Physics

Delocalization of chaotic eigenstates: an entropy approach

We focus on the localization properties of the eigenmodes of the
Laplace-Beltrami operator on compact Riemannian manifolds of (possibly
variable) negative curvature, in the high-frequency régime. Because the
geodesic flow on such a manifold is chaotic (Anosov), the Quantum Ergodicity
theorem states that "almost all" eigenmodes become equidistributed over X
in the high frequency limit; more precisely, these states equidistribute on the
phase space, that is converge to the Liouville measure.  This leaves the possibility
for sparse sequences of "exceptional eigenmodes", distributing according to different
flow-invariant measures (such measures are called semiclassical measures).
Although numerical computations indicate the existence of eigenmodes with strong
enhancements ("scars") along certain unstable closed geodesics, the Quantum
Unique Ergodicity conjecture claims that such scars are too weak to be detected
at the level of semiclassical measures, and that the unique semiclassical measure
is the Liouville measure. So far this conjecture has been proved only in the case
of arithmetic surfaces of constant negative curvature.

I will present a result "half-way" towards this QUE conjecture, in the case of
variable negative curvature. Combining methods from semiclassical analysis
and ergodic theory, we show that any semiclassical measure is at least
"half-delocalized". More precisely, we prove an explicit lower bound for the
Kolmogorov-Sinai entropies of semiclassical measures: in the case of
constant curvature, this lower bound equals half the maximal entropy.
In particular, a semiclassical measure cannot be supported on countably many
periodic geodesics. This entropic bound also applies to chaotic toy models
(quantum hyperbolic symplectomorphisms) for which QUE is known to fail,
and can even be sharp is certain cases.

(joint with Nalini Anantharaman)

Coffee will be served at 12:00 before the lecture
at Schreiber building 006