Title: Fredholm theory for scattering on asymptotically conic spaces and applications
Speaker: Andras Vasy
Speaker Info: Stanford
In this lecture I will discuss a new approach to scattering theory and some applications to the decay of waves on asymptotically flat, such as rotating Kerr black hole, spacetimes. Scattering theory in the spectral (time-independent) approach typically analyzes the near-spectrum behavior of the resolvent of a Laplacian-like operator on an asymptotically Euclidean space, including the existence of limiting resolvents in suitable function spaces. Here we set up a Fredholm theory on the spectrum, so the spectral family is inverted directly as an operator, rather than having to take limits of the resolvent.Date: Monday, May 24, 2021
Indeed, it is natural to consider geometric generalizations of the asymptotically Euclidean setting to asymptotically conic spaces $(X,g)$ of dimension at least $3$ (with the asymptotic behavior at the `large end’ of the cone). Then the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$. The results include the inversion of $P(\sigma)$ for real $\sigma\neq 0$ on function spaces that precisely encode the behavior of outgoing spherical waves, as well as the uniform description of the inverse as $\sigma\to 0$. This then provides the analytic background for the linearized black hole stability result of the author with Haefner and Hintz.