The Born approximation in the three-dimensional Calder\'on problem

Uniqueness and reconstruction in the three-dimensional Calderon inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrodinger operators $-\Delta +q $. We study the Born approximation of $q$ in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension. We show that this approximation is well-defined and obtain a closed formula that involves the spectrum of the Dirichlet-to-Neumann map associated to $-\Delta + q$. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the invariance properties of the exact inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis spherical harmonics. Motivated by these formulas we also study the high-energy behaviour of the matrix elements of the Dirichlet to Neumann map.