Quantum ergodicity: how eigenfunctions distribute in phase space
We will discuss when high-energy Laplace eigenfunctions may delocalize. Heuristically, if the corresponding (classical) geodesic flow is ergodic, then similar behavior should apply to high-energy eigenstates, i.e. they should equidistribute in a suitable sense. We will prove the Quantum Ergodicity Theorem, which states that for compact closed Riemannian manifolds with ergodic geodesic flows (in particular those of negative sectional curvature) that "most" eigenfunctions will equidistribute. The proof will consider a more refined notion of concentration, namely that in phase space, and show that most eigenfunctions will in fact equidistribute in phase space, using tools from semiclassical analysis. Time permitting, we will discuss (without too many details) the stronger Quantum Unique Ergodicity conjecture and its connection to number theory via the analysis of Maass forms.