  We study a non-relativistic particle subject to a three-dimensional spherical
potential consisting of a finite well and a radial $\delta$-$\delta'$ contact
interaction at the well edge. This contact potential is defined by appropriate
matching conditions for the radial functions, thereby fixing a self adjoint
extension of the non-singular Hamiltonian. Since this model admits exact
solutions for the wave function, we are able to characterize and calculate the
number of bound states. We also extend some well-known properties of certain
spherically symmetric potentials and describe the resonances, defined as
unstable quantum states. Based on the Woods-Saxon potential, this configuration
is implemented as a first approximation for a mean-field nuclear model. The
results derived are tested with experimental and numerical data in the double
magic nuclei $^{132}$Sn and $^{208}$Pb with an extra neutron.
