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: StÃ©phane Nonnenmacher (Institut de Physique ThÃ©orique, 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 ************************************* 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) --------------------------------------------------------- Technion Math Net-2 (TECHMATH2) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: <ostrover@post.tau.ac.il>