One of the main advantages gained by the modern use of point
interactions in mathematical physics is the exact solvability of the
Schroedinger equation and, simultaneously, the possibility of
understanding the underlying formulae simply in the language of the
essentially self-adjoint extensions. Unfortunately, the empirically
founded interpretation of real systems often prefers the
smooth-function form of the interactions. The paper shows in which
sense the two languages may be used as equivalent. More precisely, in
one dimension and for both finite and infinite numbers N of the
``points", the authors show how the mutual equivalence can be
understood in the strong resolvent sense, and also how one can keep
the large-N convergence under strict control. Noticing, in such a
context, that the recent development in the field moves towards a
weakening of the Hermiticity to mere pseudo-Hermiticity [see, e.g.,
S. Albeverio, S.M. Fei and P. Kurasov in Lett. Math. Phys 59, 227-242
(2002)], one can only hope that the rigorous studies of the presented
type will be also extended in this direction in the nearest future.