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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.
©2016 American Institute of Physics

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
26A33 Fractional derivatives and integrals
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