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

TL;DR

This work analyzes Klein–Gordon fields on asymptotically anti-de Sitter spacetimes, proving that the forward Dirichlet-to-Neumann map is a paired Lagrangian distribution that semiclassically acts as a fractional power of the boundary wave operator. It shows that, for generic mass parameters ν, the DN map determines the Taylor expansion of the bulk metric at the boundary, enabling recovery of a real-analytic or Einstein bulk metric up to isometries. A Lorentzian analogue of Graham–Zworski is established: residues of Λ_g(ν) at poles yield conformally invariant boundary operators L_k whose principal parts match powers of the boundary wave operator. The approach combines Hankel multipliers, regularized integrals, and a boundary-focused paired-Lagrangian calculus to derive both the forward map structure and inverse reconstruction results, with energy estimates extending the analysis to complex ν and an appendix addressing well-posedness in that regime.

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.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• Theorem 2.6
• Proposition 2.7