有限Kronig-Penney模型中转移矩阵的算术三角结构
Arithmetic triangular structures in the transfer-matrix of finite Kronig-Penney models
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- Università del Piemonte Orientale（皮埃蒙特东方大学）
- INFN and Centro Regge, Sezione di Torino（意大利国家核物理研究所和雷吉中心都灵分部）
- CIDMA and Departamento de Matemática, Campus Universitário de Santiago, Aveiro（阿威罗大学圣地亚哥校区数学系与CIDMA）
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中文总结 AI 辅助
本工作通过切比雪夫多项式推导有限Kronig-Penney模型转移矩阵的闭式表达式，揭示量子散射与组合结构间的对应关系。
中文摘要 AI 辅助
本工作对由Figueroa等人（2025）近期引入的一维狄拉克δ势阵列构成的有限Kronig-Penney模型的转移矩阵结构提供了完整的解析刻画。尽管他们的研究识别出了特定转移矩阵元素及相关组合系数的出现，但相应闭式表达式的严格推导尚未建立。通过将单位胞转移矩阵的N次幂表示为第二类切比雪夫多项式，我们获得了全局透射和反射振幅的显式闭式表示。所提出的公式揭示了多重量子散射过程、离散卷积模式和超复数组合结构之间先前未被认识的结构对应关系。
英文摘要
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.
