Quantum Ergodicity and beyond

4:00 pm Thursday, January 28, 2010
Stéphane Nonnenmacher (Institut de Physique Théorique, CEA-Saclay)


I will present some questions relative to quantum systems, the classical limit of which is "chaotic" (this field of research is casually called "quantum chaos"). Typical examples of such systems are given by the Laplace-Beltrami operator on a compact manifold of negative sectional curvature, or the Laplacian on some Euclidean "billiards". Specifically, I will attempt to describe the eigenmodes of such systems, in the semiclassical (or high frequency) limit. Until recently, the only general rigorous result on this question was Quantum Ergodicity, which states that almost all high frequency eigenmodes are "equidistributed" across the phase space. On the other hand, numerical investigations have revealed the possibility for some eigenmodes to concentrate near certain unstable periodic orbits (a phenomenon called "scarring"), leaving open the existence of "exceptional" localized eigenmodes. I will finally explain how the use of modern dynamical systems theory tools (namely the metric entropy) recently allowed to improve our understanding of these eigenmodes for strongly chaotic (Anosov) systems.

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