Dr Andrew Hassell (ANU), joint winner of the 2003 Australian
Mathematical Society Medal, will speak on 'Classical systems with
hyperbolic trapped sets and dispersive estimates for PDE'.
Consider the time-dependent Schrodinger equation on a complete noncompact Riemannian manifold M (for example, a manifold which looks like flat Euclidean space outside a compact set). This PDE has a dispersive character; that is, the solution cannot concentrate in a small region of space for more than a brief period of time. Various analytic estimates can be proved that give quantitative effect to this vague statement.
The precise form of these estimates depends on the dynamical properties of the associated classical system, namely geodesic flow on M (which is a Hamiltonian dynamical system). The sharpest form of the dispersive estimates are obtained when there is no trapped set, i.e. when all geodesics on the manifold M reach spatial infinity. I will talk about recent work of mine with Burq and Guillarmou, in which under suitable assumptions we can also obtain equally sharp estimates when trapping is present. The most important assumption is that the trapped set is hyperbolic (unstable).