Reconstructions from boundary measurements on admissible manifolds

Abstract

We prove that a potential q can be reconstructed from the Dirichletto-Neumann map for the Schrödinger operator -Δg + q in a fixed admissible 3-dimensional Riemannian manifold (M; g). We also show that an admissible metric g in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for Δg. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry. © 2011 American Institute of Mathematical Sciences.