The Calderón problem is an inverse source problem

Abstract

We prove that uniqueness for the Calderón problem on a Riemannian manifold with boundary follows from a hypothetical unique continuation property for the elliptic operator $\Delta +V+({\Lambda }_t^1-q)\otimes ({\Lambda }_t^2-q)$ defined on $\partial {\mathcal{M}}^2\times [0,1]$ where $V$ and $q$ are potentials and ${\Lambda }_t^i$ is a Dirichlet-Neumann operator at depth $t$. This is done by showing that the difference of two Dirichlet-Neumann maps is equal to the Neumann boundary values of the solution to an inhomogeneous equation for said operator, where the source term is a measure supported on the diagonal of $\partial {\mathcal{M}}^2$.