On fractional semidiscrete Dirac operators of L\'evy-Leblond type

Abstract

In this paper we introduce a wide class of space-fractional and time-fractional semidiscrete Dirac operators of L\'evy-Leblond type on the semidiscrete space-time lattice $h\mathbb{Z}^n\times[0,\infty)$ ($h>0$), resembling to fractional semidiscrete counterparts of the so-called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup $\left\{\exp(-te^{i\theta}(-\Delta_h)^{\alpha})\right\}_{t\geq 0}$, carrying the parameter constraints $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha \pi}{2}$. The results obtained involve the study of Cauchy problems on $h\mathbb{Z}^n\times[0,\infty)$.