Quantum Ergodicity and the Analysis of Semiclassical Pseudodifferential Operators

TL;DR

This thesis explores quantum ergodicity and unique ergodicity on Riemannian manifolds, providing an elementary geometric framework, summarizing key proofs, and illustrating concepts through simulations of billiard flows.

Contribution

It offers an accessible geometric and analytic perspective on quantum ergodicity, including detailed explanations of proofs and visualizations of quantum chaos phenomena.

Findings

Eigenfunctions equidistribute in phase space under negative curvature.

Quantum unique ergodicity implies no scarring in eigenfunctions.

Graphical simulations illustrate billiard flow behaviors.

Abstract

This undergraduate thesis is concerned with developing the tools of differential geometry and semiclassical analysis needed to understand the the quantum ergodicity theorem of Schnirelman (1974), Zelditch (1987), and Colin de Verdi\`ere (1985) and the quantum unique ergodicity conjecture of Rudnick and Sarnak (1994). The former states that, on any Riemannian manifold with negative curvature or ergodic geodesic flow, the eigenfunctions of the Laplace-Beltrami operator equidistribute in phase space with density 1. Under the same assumptions, the latter states that the eigenfunctions induce a sequence of Wigner probability measures on fibers of the Hamiltonian in phase space, and these measures converge in the weak-* topology to the uniform Liouville measure. If true, the conjecture implies that such eigenfunctions equidistribute in the high-eigenvalue limit with no exceptional "scarring" patterns. This physically means that the finest details of chaotic Hamiltonian systems can never reflect their quantum-mechanical behaviors, even in the semiclassical limit. The main contribution of this thesis is to contextualize the question of quantum ergodicity and quantum unique ergodicity in an elementary analytic and geometric framework. In addition to presenting and summarizing numerous important proofs, such as Colin de Verdi\`ere's proof of the quantum ergodicity theorem, we perform graphical simulations of certain billiard flows and expositorily discuss several themes in the study of quantum chaos.