Summary: ENTROPY OF EIGENFUNCTIONS
NALINI ANANTHARAMAN, HERBERT KOCH, AND STÉPHANE NONNENMACHER
Abstract. We study the high--energy limit for eigenfunctions of the laplacian, on a
compact negatively curved manifold. We review the recent result of Anantharaman--
Nonnenmacher [4] giving a lower bound on the Kolmogorov--Sinai entropy of semiclassical
measures. The bound proved here improves the result of [4] in the case of variable negative
curvature.
1. Motivations
The theory of quantum chaos tries to understand how the chaotic behaviour of a classi­
cal Hamiltonian system is reflected in its quantum counterpart. For instance, let M be a
compact Riemannian C # manifold, with negative sectional curvatures. The geodesic flow
has the Anosov property, which is considered as the ideal chaotic behaviour in the theory
of dynamical systems. The corresponding quantum dynamics is the unitary flow gener­
ated by the Laplace­Beltrami operator on L 2 (M). One expects that the chaotic properties
of the geodesic flow influence the spectral theory of the Laplacian. The Random Matrix
conjecture [7] asserts that the large eigenvalues should, after proper unfolding, statisti­
cally resemble those of a large random matrix, at least for a generic Anosov metric. The
Quantum Unique Ergodicity conjecture [26] (see also [6, 30]) describes the corresponding
eigenfunctions # k : it claims that the probability measure |# k (x)| 2 dx should approach (in
the weak topology) the Riemannian volume, when the eigenvalue tends to infinity. In fact

Source: Anantharaman, Nalini - Centre de Mathématiques Laurent Schwartz, École Polytechnique

Collections: Mathematics