The linearized Calderon problem in transversally anisotropic geometries

TL;DR

This paper investigates the linearized anisotropic Calderon problem on transversally anisotropic manifolds, showing boundary measurements determine a transform that recovers transversal singularities without relying on the geodesic X-ray transform.

Contribution

It introduces a new approach to the linearized Calderon problem using an FBI type transform, bypassing limitations of previous geodesic X-ray transform methods.

Findings

Boundary measurements determine an FBI type transform in the transversal manifold.

The method recovers transversal singularities in the linearized problem.

A geometric condition on the transversal manifold is required for the results.

Abstract

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem.