Direct and inverse scattering on conformally compact manifolds and asymptotically hyperbolic manifolds with polyhomogeneous metric

Leonardo Marazzi, Purdue University

Abstract

We present two kinds of results, namely three inverse theorems and a spectral result. The first inverse theorem we prove is that the scattering matrix of Δg + V, g conformally compact, V ∈ C∞, at a fixed energies (λ1, λ2) in a suitable subset of [special characters omitted], determines the curvature at infinity α( y), and the Taylor series of both the potential and the metric g at the boundary. We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion at the boundary. We prove an inverse theorem in this setting and use it to prove an inverse theorem for asymptotically hyperbolic odd dimensional Einstein manifolds. The latter relates a modified scattering operator we define to the geometry of the manifold. We also prove that the resolvent always has essential singularities.

Degree

Ph.D.

Advisors

Barreto, Purdue University.

Subject Area

Mathematics

COinS