The Calder\'on problem is an inverse source problem

Abstract

We prove that uniqueness for the Calder\'on problem on a Riemannian manifold with boundary follows from a hypothetical unique continuation property for the elliptic operator $\Delta+V+(\Lambda^{1}_{t}-q)\otimes (\Lambda^{2}_{t}-q)$ defined on $\partial\mathcal{M}^{2}\times [0,1]$ where $V$ and $q$ are potentials and $\Lambda^{i}_{t}$ 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 $\partial\mathcal{M}^{2}$.