Reconstructions from boundary measurements on admissible manifolds

Abstract

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-{\Delta }_g+q$ 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 ${\Delta }_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. on admissible manifolds, and extends the reconstruction procedure of Nachman 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.

19 pages