Mode localization on chaotic manifolds: an entropy approach (Collab. with N.Anantharaman)

Stephane Nonnenmacher
Service de Physique Theorique CEN Saclay

We theoretically investigate the structure of high-energy eigenmodes of the
Laplace-Beltrami operator on compact Riemannian manifolds, the geodesic flow of which is chaotic (Anosov). This is for instance the case if the sectional curvature is negative everywhere.
In the high-energy limit, the (phase) space distribution of eigenmodes can be described
by "semiclassical measures", which are probability measures on the unit cotangent bundle,
invariant by the geodesic flow. Using the metric entropy as an indicator of localization, we show
that high-energy eigenmodes are "at least half-delocalized". More precisely, the entropy of
semiclassical measures is at least equal to half the maximal entropy. This result rules out the existence of "strong scars"
(eigenstates localized along classical periodic geodesics), and is a step towards the "quantum unique ergodicity" conjecture for such manifolds.

Presentation (PDF File)

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