Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

TL;DR

This paper investigates the localization properties of high-energy eigenfunctions of the Laplacian on Anosov manifolds, revealing they are at least half-delocalized due to entropy bounds related to their semiclassical measures.

Contribution

It establishes a lower bound on the Kolmogorov-Sinai entropy of semiclassical measures for eigenfunctions on Anosov manifolds, showing partial delocalization at high energies.

Findings

Entropy of semiclassical measures is bounded below by a positive constant.

High-energy eigenfunctions are at least half-delocalized.

In constant negative curvature, entropy bound equals half the maximum.

Abstract

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai 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, high-energy eigenfunctions are at least half-delocalized.