Conditions for realizing one-point interactions from a multi-layer structure model

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.