School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

Institut de Physique Théorique, CEA-Saclay

TIDY Distinguished lecture in Mathematical Physics

We focus on the localization properties of the eigenmodes of the

Abstract:

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