  In this article we prove the existence of the Born approximation in the
context of the radial Calder\'on problem for Schr\"odinger operators. This is
the inverse problem of recovering a radial potential on the unit ball from the
knowledge of the Dirichlet-to-Neumann map (DtN map from now on) of the
corresponding Schr\"odinger operator. The Born approximation naturally appears
as the linear component of a factorization of the Calder\'on problem; we show
that the non-linear part, obtaining the potential from the Born approximation,
enjoys several interesting properties. First, this map is local, in the sense
that knowledge of the Born approximation in a neighborhood of the boundary is
equivalent to knowledge of the potential in the same neighborhood, and, second,
it is H\"older stable. This shows in particular that the ill-posedness of the
Calder\'on problem arises solely from the linear step, which consists in
computing the Born approximation from the DtN map by solving a Hausdorff moment
problem. Moreover, we present an effective algorithm to compute the potential
from the Born approximation and show a result on reconstruction of
singularities. Finally, we use the Born approximation to obtain a partial
characterization of the set of DtN maps for radial potentials. The proofs of
these results do not make use of Complex Geometric Optics solutions or its
analogues; they are based on results on inverse spectral theory for
Schr\"odinger operators on the half-line, in particular on the concept of
$A$-amplitude introduced by Barry Simon.
