Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models
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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BibTeX
@misc{aceto2026arithmetic,
title = {{Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models}},
author = {Lidia Aceto and Pietro Antonio Grassi and Helmuth Robert Malonek},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08523},
eprint = {2609.08523},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08523},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models
AU - Lidia Aceto
AU - Pietro Antonio Grassi
AU - Helmuth Robert Malonek
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08523
UR - https://arxiv.org/abs/2609.08523
AB - 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.
ER -