Definition of the Riesz derivative and its application to space fractional quantum mechanics

We investigate and compare different representations of the Riesz derivative, which plays an important role in anomalous diffusion and space fractional quantum mechanics. In particular, we show that a certain representation of the Riesz derivative, Rxα, that is generally given as also valid for α = 1, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the α → 1 limit of the space fractional quantum mechanics and its consistency.