The anisotropic Calder\'on problem at large fixed frequency on manifolds with invertible ray transform

TL;DR

This paper addresses the inverse problem of recovering potentials on Riemannian manifolds using boundary measurements at fixed high frequency, extending previous results to more general manifolds with invertible ray transforms.

Contribution

It generalizes the Calderón problem to manifolds with stably invertible geodesic ray transforms, employing Gaussian beam quasimodes with uniform bounds.

Findings

Extended Calderón problem results to manifolds with invertible ray transform

Constructed Gaussian beam quasimodes with uniform bounds

Demonstrated stable potential recovery at large fixed frequency

Abstract

We consider the inverse problem of recovering a potential from the Dirichlet to Neumann map at a large fixed frequency on certain Riemannian manifolds. We extend the earlier result of [G. Uhlmann and Y. Wang, arXiv:2104.03477] to the case of simple manifolds, and more generally to manifolds where the geodesic ray transform is stably invertible. The argument involves an invariantly formulated construction of Gaussian beam quasimodes with uniform bounds for the underlying constants.