  Uniqueness and reconstruction in the three-dimensional Calder\'on inverse
conductivity problem can be reduced to the study of the inverse boundary
problem for Schr\"odinger 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 $d\ge 3$. We show that this approximation satisfies
a closed formula that only involves the spectrum of the Dirichlet-to-Neumann
map associated to $-\Delta + q$, which is closely related to a particular
moment problem. We then turn to general real and essentially bounded potentials
in three dimensions and introduce the notion of averaged Born approximation,
which captures the exact invariance properties of the 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. To show that the averaged Born approximation does not destroy
information on the potential, we also study the high-energy behavior of the
matrix elements of the Dirichlet to Neumann map.
