Quantum Analogs of Fractional Derivatives

Abstract

The fractional derivative has different definitions (Samko et al., 1993; Kubas et al., 2006), and exploiting any of them depends on the kind of the problems, initial (boundary) conditions, and the specifics of the considered physical processes. The classical definitions are the so-called Riemann-Liouville and Liouville derivatives (Kubas et al., 2006). These fractional derivatives are defined by the same equations on a finite interval of ℝ and of the real axis ℝ, correspondently. Note that the Caputo and Riesz derivatives can be represented (Kubas et al., 2006; Samko et al., 1993) through the Riemann-Liouville and Liouville derivatives. Therefore quantization of Riemann-Liouville and Liouville fractional derivatives allows us to derive quantum analogs for Caputo and Riesz derivatives.