Entropy of chaotic eigenstates

TL;DR

This paper investigates the entropy properties of high-frequency eigenstates of the Laplacian on negatively curved manifolds, establishing lower bounds on their semiclassical measures to show they cannot be overly localized.

Contribution

It provides new lower bounds on the Kolmogorov-Sinai entropy of semiclassical measures for eigenstates, extending results to Anosov flows and diffeomorphisms.

Findings

High-frequency eigenstates have a positive lower bound on entropy.

Eigenstates cannot be too localized in phase space.

Results extend to classical Anosov systems and quantized diffeomorphisms.

Abstract

These notes present a recent approach to study the high-frequency eigenstates of the Laplacian on compact Riemannian manifolds of negative sectional curvature. The main result is a lower bound on the Kolmogorov-Sinai entropy of the semiclassical measures associated with sequences of eigenstates, showing that high-frequency eigenstates cannot be too localized. The method is extended to the case of semiclassical Hamiltonian operators for which the classical flow in some energy range is of Anosov type, and to the case of quantized Anosov diffeomorphisms on the torus.