  In this paper we give an application of Dalgarno-Lewis method, the latter not
usually taught in quantum mechanics courses. This is very unfortunate since
this method allows to bypass the sum over states appearing in the usual
perturbation theory. In this context, and as an example, we study the effect of
an external field, both static and frequency dependent, on a model-atom at
fixed distance from a substrate. This can happen, for instance, when some
organic molecule binds from one side to the substrate and from the other side
to an atom or any other polarizable system. We model the polarizable atom by a
short range potential, a Dirac$-\delta$ and find that the existence of a bound
state depends on the ratio of the effective "nuclear charge" to the distance of
the atom to the substrate. Using an asymptotic analysis, previously developed
in the context of a single $\delta-$function potential in an infinite medium,
we determine the ionization rate and the Stark shift of our system. Using
Dalgarno-Lewis theory we find an exact expression for the static and dynamic
polarizabilities of our system valid to all distances. We show that the
polarizability is extremely sensitive to the distance to the substrate creating
the possibility of using this quantity as a nanometric ruler. Furthermore, the
line shape of the dynamic polarizability is also extremely sensitive to the
distance to the substrate, thus providing another route to measure nanometric
distances. The ditactic value of the $\delta-$function potential is well
accepted in teaching activities due to its simplicity, while keeping the
essential ingredients of a given problem.
