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

Abstract

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta _+^{(\alpha , \beta , \gamma )}:= D_{x_0^+}^{1+\alpha } +D_{y_0^+}^{1+\beta } +D_{z_0^+}^{1+\gamma },$ where $(\alpha , \beta , \gamma ) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha }, D_{y_0^+}^{1+\beta }, D_{z_0^+}^{1+\gamma }$ 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 $\Delta _+^{(\alpha ,\beta ,\gamma )}$ 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.