  A heterostructure composed of $N$ parallel homogeneous layers is studied in
the limit as their widths $l_1, \ldots , l_N$ shrink to zero. The problem is
investigated in one dimension and the piecewise constant potential in the
Schr\"{o}dinger equation is given by the strengths $V_1, \ldots , V_N$ as
functions of $l_1, \ldots , l_N$, respectively. The key point is the derivation
of the conditions on the functions $V_1(l_1), \ldots , V_N(l_N)$ for realizing
a family of one-point interactions as $l_1, \ldots , l_N$ tend to zero along
available paths in the $N$-dimensional space. The existence of equations for a
squeezed structure, the solution of which determines the system parameter
values, under which the non-zero tunneling of quantum particles through a
multi-layer structure occurs, is shown to exist and depend on the paths. This
tunneling appears as a result of an appropriate cancellation of divergences.
