Tel Aviv University, Colloquium
 
Hello everyone,
 
The next colloquium talk will be held on:
 
Monday, 7/01/2013, 12:15, Schreiber 006, Tel Aviv University.
 
Speaker: Stéphane Nonnenmacher (Institut de Physique Théorique,
CEA-Saclay)
 
Title: Delocalization of chaotic eigenstates: an entropy approach.
 
The abstract is given below. Tea and coffee at 12:00, same room.
 
Hope to see you there. For information about future colloquia, see
 <http://www.math.tau.ac.il/~ostrover/colloquium/colloq2012.html>
 
Yours,
Yaron
 
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Abstract: 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 regime.
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)
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from:  <ostrover@post.tau.ac.il>