The Dirichlet-to-Neumann map on asymptotically anti-de Sitter spaces and holography

Abstract

We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes, and show that the forward Dirichlet-to-Neumann map (or scattering matrix) is a fractional power of the boundary wave operator modulo lower order terms in the sense of paired Lagrangian distributions. We use it to show that, outside of a countable set of mass parameters, the Dirichlet-to-Neumann map determines the Taylor series of the bulk metric at the boundary, and hence allows the recovery of a real analytic metric or Einstein metric modulo isometries. Furthermore, we prove a Lorentzian version of the Graham-Zworski theorem relating poles of the Dirichlet-to-Neumann map to conformally invariant powers of the boundary wave operator.