AI 中文总结
该研究在(N+1)维Clifford分析中，通过Clifford-Appell多项式基将狄拉克型Kronig-Penney模型的微分问题转化为代数递推，建立了自由传播问题的统一Clifford分析公式化及与广义亥姆霍兹方程的联系。
AI 中文摘要
我们在(N+1)维Clifford分析中研究与广义Kronig-Penney模型相关的定态狄拉克方程。在相互作用位点外，通过引入合适的广义柯西-黎曼算子重新表述自由狄拉克系统，得到两个Clifford值场的耦合一阶方程组。利用特征超复变量，该方程组被约化为一对解耦的广义亥姆霍兹方程。为构造显式解，我们引入适配于广义柯西-黎曼算子的二元Clifford-Appell多项式基。Appell展开将微分问题转化为展开系数的代数递推关系，得到解的显式级数表示。所提框架为自由传播问题提供了统一的Clifford分析公式化，并建立了广义亥姆霍兹方程与超复函数论中Appell多项式系统的显式联系。
英文摘要
We investigate the stationary Dirac equation associated with a generalized Kronig-Penney model in (N+1)-dimensional Clifford analysis. Away from the interaction sites, the free Dirac system is reformulated by introducing suitable generalized Cauchy-Riemann operators, leading to a coupled first-order system for two Clifford-valued fields. By means of characteristic hypercomplex variables, this system is reduced to a pair of decoupled generalized Helmholtz equations. To construct explicit solutions, we introduce a bivariate Clifford-Appell polynomial basis adapted to the generalized Cauchy-Riemann operators. The Appell expansions transform the differential problem into an algebraic recurrence for the expansion coefficients, yielding explicit series representations of the solutions. The proposed framework provides a unified Clifford-analytic formulation of the free propagation problem and establishes an explicit connection between generalized Helmholtz equations, and Appell polynomial systems in hypercomplex function theory.
CommentsPages 24, no figure
