Summary: HALF­DELOCALIZATION OF EIGENFUNCTIONS FOR THE
LAPLACIAN ON AN ANOSOV MANIFOLD
NALINI ANANTHARAMAN AND STÉPHANE NONNENMACHER
Abstract. We study the high­energy eigenfunctions of the Laplacian on a compact Rie­
mannian manifold with Anosov geodesic flow. The localization of a semiclassical measure
associated with a sequence of eigenfunctions is characterized by the Kolmogorov­Sinai
entropy of this measure. We show that this entropy is necessarily bounded from below
by a constant which, in the case of constant negative curvature, equals half the maximal
entropy. In this sense, high­energy eigenfunctions are at least half­delocalized.
The theory of quantum chaos tries to understand how the chaotic behaviour of a clas­
sical Hamiltonian system is reflected in its quantum version. For instance, let M be a
compact Riemannian C # manifold, such that the geodesic flow has the Anosov property
--- the ideal chaotic behaviour. The corresponding quantum dynamics is the unitary flow
generated 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 [6] asserts that the high­lying eigenvalues should, after proper renormal­
ization, statistically resemble those of a large random matrix, at least for a generic Anosov
metric. The Quantum Unique Ergodicity conjecture [27] (see also [5, 30]) deals with the
corresponding eigenfunctions #: it claims that the probability density |#(x)| 2 dx should
approach (in a weak sense) the Riemannian volume, when the eigenvalue corresponding

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

Collections: Mathematics