Michael Taylor, UNC Chapel Hill – Analysis & PDE Seminar

Title: Variations on quantum ergodic theorems

Abstract: Free quantum motion on a compact Riemannian manifold M is described by the energy levels and eigenfunctions of the Laplace-Beltrami operator.  A lot of work studies the connection between the behavior of the geodesic flow on M and the quantum data.  Quantum ergodic theorems draw conclusions from the long time behavior of the geodesic flow.  Frequently they make the hypothesis that the geodesic flow is ergodic, but of course many geodesic flows are not ergodic.  We discuss formulations of quantum ergodic theorems that have interesting conclusions regardless of whether the geodesic flow is ergodic.  We explain how such results naturally motivate one to examine quantization of non-smooth symbols.

Time permitting, we also mention a recent DMJ article by Colin de Verdiere et al., dealing with microlocal concentration of eigenfunctions of a class of subelliptic operators (of a sort first studied for their connection to the dbar-Neumann problem).  We discuss how one can make a finer microlocalization than done in that paper.  This finer study involves a “noncommutative microlocal analysis.”