On fractional semidiscrete Dirac operators of Lévy-Leblond type

Abstract

In this paper we introduce a wide class of space-fractional and time-fractional semidiscrete Dirac operators of Lévy-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 ${\left\{\exp (-t{e}^{i\theta }(-{\Delta }_h{)}^{\alpha })\right\}}_{t\ge 0}$, carrying the parameter constraints $0<\alpha \le 1$ and $|\theta |\le \frac{\alpha \pi }{2}$. The results obtained involve the study of Cauchy problems on $h{\mathbb{Z}}^n\times [0,\infty )$.

30 pages, 3 figures; several typos were corrected; Section 5 was expanded

References in corpus (4)

• The fundamental solution of the space-time fractional diffusion equation
• Brownian subordinators and fractional Cauchy problems
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie Symmetries of the Lévy-Leblond Equations
• Solutions for the Lévy-Leblond or parabolic Dirac equation and its generalizations

Cited by in corpus (1)

• Graded colour Lie superalgebras for solving Lévy-Leblond equations