Inverse scattering at fixed energy in de-Sitter-Reissner-Nordstrom black holes.

In this talk, we shall consider massless Dirac fields evolving in the outer region of de-Sitter-Reissner-Nordstrom Black Holes, a class of spherically symmetric, electrically charged, spacetimes with positive cosmological constant. We shall first define the corresponding partial wave scattering matrices S(k,n), objects that encode the scattering properties of an incoming Dirac waves having energy k and angular momentum n. We shall then show that the metric of such black holes is uniquely determined by the knowledge of the partial scattering matrices at a fixed non-zero energy and for almost all angular momenta. The main tool used to prove this result consists in complexifying the angular momentum and in using the particular analytic properties of the unphysical scattering matrix S(k,z) where z belongs now to the complex plane. Ths result was obtained in collaboration with François Nicoleau (Université de Nantes).