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Definition of the Riesz derivative and its application to space fractional quantum mechanics. (English) Zbl 1353.81041
Summary: 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, $$R_x^{\alpha}$$, that is generally given as also valid for $$\alpha = 1$$, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the $$\alpha \to 1$$ limit of the space fractional quantum mechanics and its consistency.