The Calder\'on problem in the $L^p$ framework on Riemann surfaces

TL;DR

This paper extends the uniqueness results of the Calderón problem on Riemann surfaces to unbounded potentials in $L^p$, using complex geometric optics and Carleman estimates to recover potentials from Cauchy data.

Contribution

It introduces a method to determine unbounded potentials in $L^p$ on Riemann surfaces from boundary measurements, expanding previous results to more general potentials.

Findings

Uniqueness of potential recovery in $L^p$ for $p>4/3$

Recovery of potential singularities from Cauchy data

Application of stationary phase method in this context

Abstract

The purpose of this article is to extend the uniqueness results for the two dimensional Calder\'on problem to unbounded potentials on general geometric settings. We prove that the Cauchy data sets for Schr\"odinger equations uniquely determines potentials in $L^{p}$ for $p> 4/3$. In doing so, we first recover singularities of the potential, from which point a $L^2$-based method of stationary phase can be applied. Both steps are done via constructions of complex geometric optic solutions and Carleman estimates.