Reconstructions from boundary measurements on admissible manifolds

TL;DR

This paper demonstrates how to reconstruct a potential and metric on admissible 3D manifolds from boundary measurements, extending previous uniqueness results with constructive methods using boundary integral equations and layer potentials.

Contribution

It provides a constructive reconstruction method for potentials and metrics on admissible manifolds from boundary data, extending prior uniqueness results with explicit procedures.

Findings

Reconstruction of potential q from Dirichlet-to-Neumann map.

Construction of admissible metric g from boundary data.

Development of boundary integral equations and layer potentials.

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.