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

• URL: http://arxiv.org/abs/2112.08510v1
• Date: Wed, 15 Dec 2021 22:30:39 GMT
• Title: Conditions for realizing one-point interactions from a multi-layer structure model
• Authors: Alexander V. Zolotaryuk and Yaroslav Zolotaryuk
• Abstract summary: A heterostructure composed of $N$ parallel homogeneous layers is studied in the limit as their widths shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schr"odinger equation is given.
• Score: 77.34726150561087
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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