Determining an unbounded potential from Cauchy data in admissible geometries

TL;DR

This paper extends the uniqueness results for inverse Schrödinger problems in admissible geometries to include unbounded potentials in L^{n/2}, using new Carleman estimates with limiting weights.

Contribution

It introduces a method to determine unbounded potentials in inverse problems on certain manifolds, expanding prior results that only covered bounded potentials.

Findings

Uniqueness of potential determination extended to unbounded cases in L^{n/2}.

Derived new Lp Carleman estimates with limiting weights.

Generalized inverse problem techniques to broader class of potentials.

Abstract

In a previous article of Dos Santos Ferreira, Kenig, Salo and Uhlmann, anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schr\"odinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n \geq 3. In this article we extend this result to the case of unbounded potentials, namely those in Ln/2. In the process, we derive Lp Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison-Kenig and Kenig-Ruiz-Sogge.