Reconstruction in the Calder\'on problem on conformally transversally anisotropic manifolds

TL;DR

This paper provides a constructive method to determine a potential in a Schrödinger operator on certain manifolds from boundary measurements, extending previous uniqueness results with explicit reconstruction techniques.

Contribution

It introduces a constructive reconstruction procedure for the potential in the Calderón problem on conformally transversally anisotropic manifolds, utilizing Gaussian beams and boundary trace analysis.

Findings

Constructive determination of potential from Dirichlet-to-Neumann map.

Use of Gaussian beams for boundary trace reconstruction.

Identification of boundary space with standard Sobolev space.

Abstract

We show that a continuous potential $q$ can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the Schr\"odinger operator $-\Delta_g+q$ on a conformally transversally anisotropic manifold of dimension $\geq 3$, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of Dos Santos Ferreira-Kurylev-Lassas-Salo. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of Nachman-Street to our setting. We also identify the main space introduced by Nachman-Street with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet-to-Neumann map.