Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case - Ferreira, Milton (2019)
In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $Δ_+^{(α,β,γ)}:= D_{x_0^+}^{1+α} +D_{y_0^+}^{1+β} +D_{z_0^+}^{1+γ},$ where $(α, β, γ) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+α}$, $D_{y_0^+}^{1+β}$, $D_{z_0^+}^{1+γ}$ 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 $Δ_+^{(α,β,γ)}$ 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.