Quantum localization and delocalization

Quantum mechanics is described through solutions to the Schrödinger equation, which is the quantum analog of the Hamiltonian equations in classical dynamics. Stationary states are given by eigenfunctions of an elliptic operator, which in this talk will be the Laplace-Beltrami operator in a compact Riemannian manifold. Its eigenfunctions that correspond to very large eigenvalues are somewhat described by the Hamiltonian classical dynamics, which in our case is the geodesic flow. The nature of this correspondence is elusive, and depends on global geometric properties of the manifold. We will try to introduce the audience to this area of research through several simple, yet important examples, focusing mainly on manifolds with completely integrable geometry and their perturbations.