Eigenfunctions and Fundamental Solutions of the Fractional Laplace and Dirac Operators: The Riemann-Liouville Case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator Δ+(α,β,γ):=Dx0+1+α+Dy0+1+β+Dz0+1+γ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _+^{(\alpha , \beta , \gamma )}:= D_{x_0^+}^{1+\alpha } +D_{y_0^+}^{1+\beta } +D_{z_0^+}^{1+\gamma },$$\end{document} where (α,β,γ)∈]0,1]3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , \beta , \gamma ) \in \,]0,1]^3$$\end{document}, and the fractional derivatives Dx0+1+α,Dy0+1+β,Dz0+1+γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{x_0^+}^{1+\alpha }, D_{y_0^+}^{1+\beta }, D_{z_0^+}^{1+\gamma }$$\end{document} are in the Riemann–Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator Δ+(α,β,γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _+^{(\alpha ,\beta ,\gamma )}$$\end{document} in classes of functions admitting a summable fractional derivative. Making use of the Mittag–Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.