Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography

We consider a boundary value problem for the Schrödinger operator $- \Delta + q(x)$ in a ball $\Omega : (x_1 + R)^2 + x_2^2 + (x_3 - r)^2 < r^2$, whose boundary we regard as a horosphere in the hyperbolic space $ H^3$ realized in the upper half space $ R^3_+$. Let $S = \{|x| = R, x_3 > 0\}$ be a hemisphere, which is generated by a family of geodesics in $ H^3$. By imposing a suitable boundary condition on $\partial\Omega$ in terms of a pseudo-differential operator, we compute the integral mean of $q(x)$ over $S\cap\Omega$ from the local knowledge of the associated (generalized) Robin-to-Dirichlet map for $- \Delta + q(x)$ around $S\cap\partial\Omega$. The potential $q(x)$ is then reconstructed by virtue of the inverse Radon transform on hyperbolic space. If the support of $q(x)$ has a positive distance from $\partial\Omega$, one can construct this generalized Robin-to-Dirichlet map from the usual Dirichlet-to-Neumann map. These results explain the mathematical background of the well-known Barber-Brown algorithm in electrical impedance tomography.