Entropy of logarithmic modes

TL;DR

This paper investigates the entropy of semiclassical measures associated with quasimodes on negatively curved manifolds, establishing a lower bound that depends explicitly on a spectral parameter, advancing understanding of quantum chaos.

Contribution

It provides a new lower bound for the Kolmogorov-Sinai entropy of semiclassical measures in a critical spectral regime on Anosov manifolds.

Findings

Derived explicit entropy lower bounds depending on spectral width parameter.

Extended previous results to a critical delocalization regime.

Connected entropy bounds with quantum unique ergodicity conjectures.

Abstract

Let $(M,g)$ be a compact, boundaryless, Riemannian manifold whose geodesic flow on its unit sphere bundle is Anosov. Consider the (semiclassical) Laplace-Beltrami operator on $M$. Let $\epsilon >0$. We study the semiclassical measures $\mu_{sc}$ of quasimodes spectrally supported in intervals of width $\epsilon \frac{h}{|\log h|}$, a critical-type regime when considering ``delocalization". We derive a lower bound for the Kolmogorov-Sinai entropy of $\mu_{sc}$ that depends explicitly on $\epsilon$, in the spirit of that given by Ananthamaran-Koch-Nonnenmacher.