  We consider non-linear Schr\"odinger equations with a potential, and
non-local non-linearities, that are models in mesoscopic physics, for example
of a quantum capacitor, and that also are models of molecular structure. We
study in detail the initial value problem for these equations. In particular,
existence and uniqueness of local and global solutions, continuous dependence
on the initial data and regularity. We allow for a large class of unbounded
potentials. We have no restriction on the growth at infinity of the positive
part of the potential. We also construct the scattering operator in the case of
potentials that go to zero at infinity. Furthermore, we give a method for the
unique reconstruction of the potential from the small amplitude limit of the
scattering operator. In the case of the quantum capacitor, our method allows us
to uniquely reconstruct all the physical parameters from the small amplitude
limit of the scattering operator.
