Clifford-Appell formulation of a Dirac-type Kronig-Penney model in condensed matter physics

Abstract

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.