  In two and three dimensions, the standard treatment of the scattering problem
for a multi-delta-function potential,
$v(\mathbf{r})=\sum_{n=1}^N\mathfrak{z}_n\delta(\mathbf{r}-\mathbf{a}_n)$,
leads to divergent terms. Regularization of these terms and renormalization of
the coupling constants $\mathfrak{z}_n$ give rise to a finite expression for
the scattering amplitude of this potential, but this expression has an
important short-coming; in the limit where the centers $\mathbf{a}_n$ of the
delta functions coincide, it does not reproduce the formula for the scattering
amplitude of a single-delta-function potential, i.e., it seems to have a wrong
coincidence limit. We provide a critical assessment of the standard treatment
of these potentials and offer a resolution of its coincidence-limit problem.
This reveals some previously unnoticed features of this treatment. For example,
it turns out that the standard treatment is incapable of determining the
dependence of the scattering amplitude on the distances between the centers of
the delta functions. This is in sharp contrast to the treatment of this problem
offered by a recently proposed dynamical formulation of stationary scattering.
For cases where the centers of the delta functions lie on a straight line, this
formulation avoids singularities of the standard approach and yields an
expression for the scattering amplitude which has the correct coincidence
limit.
