  We investigate the eigenvalues of perturbed spherical Schr\"odinger operators
under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in
L^1(0,1)$. We show that the square roots of eigenvalues are given by the square
roots of the unperturbed eigenvalues up to an decaying error depending on the
behavior of $q(x)$ near $x=0$. Furthermore, we provide sets of spectral data
which uniquely determine $q(x)$.
