Linearized Calder\'on problem and exponentially accurate quasimodes for analytic manifolds

TL;DR

This paper solves the linearized anisotropic Calderón problem on certain analytic manifolds by constructing exponentially accurate Gaussian beam quasimodes and analyzing the analytic wave front set, without relying on geodesic X-ray transform injectivity.

Contribution

It introduces a method to solve the linearized Calderón problem on transversally analytic manifolds using Gaussian beam quasimodes and FBI transform techniques.

Findings

Successfully solves the linearized anisotropic Calderón problem under specific geometric conditions.

Constructs Gaussian beam quasimodes with exponentially small errors.

Provides a new approach that does not depend on geodesic X-ray transform injectivity.

Abstract

In this article we study the linearized anisotropic Calder\'on problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calder\'on problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.