Macroscopic limits of chaotic eigenfunctions

TL;DR

This paper reviews the relationship between high-energy eigenfunctions of the Laplacian on compact manifolds and the dynamics of geodesic flow, highlighting key results and conjectures in quantum chaos using microlocal analysis.

Contribution

It provides an overview of quantum ergodicity, quantum unique ergodicity, entropy bounds, and discusses quantum cat maps as a toy model for quantum chaos.

Findings

Quantum ergodicity theorem and its implications

Quantum unique ergodicity conjecture and challenges

Entropy bounds and eigenfunction mass distribution

Abstract

We give an overview of the interplay between the behavior of high energy eigenfunctions of the Laplacian on a compact Riemannian manifold and the dynamical properties of the geodesic flow on that manifold. This includes the Quantum Ergodicity theorem, the Quantum Unique Ergodicity conjecture, entropy bounds, and uniform lower bounds on mass of eigenfunctions. The above results belong to the domain of quantum chaos and use microlocal analysis, which is a theory behind the classical/quantum, or particle/wave, correspondence in physics. We also discuss the toy model of quantum cat maps and the challenges it poses for Quantum Unique Ergodicity.