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arXiv · 2609.08523

Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models

Lidia Aceto·Pietro Antonio Grassi·Helmuth Robert Malonek
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Abstract

This work provides a complete analytical characterization of the transfer- matrix structure associated with the finite Kronig-Penney model consisting of one-dimensional arrays of Dirac delta potentials recently introduced by Figueroa et al. (2025). Although their study identified the emergence of specific transfer-matrix entries and related combinatorial coefficients, a rig- orous derivation of the corresponding closed-form expressions has not yet been established. By expressing the N th power of the unit-cell transfer ma- trix in terms of Chebyshev polynomials of the second kind, we obtain explicit closed-form representations for the global transmission and reflection ampli- tudes. The proposed formulation reveals a previously unrecognized structural correspondence between multiple quantum scattering processes, discrete con- volutional patterns, and hypercomplex combinatorial structures.

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Lidia Aceto, Pietro Antonio Grassi, Helmuth Robert Malonek. 2026-09-08. Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models. https://arxiv.org/abs/2609.08523

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