# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Oldham, K. B.; Spanier, J., The Fractional Calculus, (1974), Dover · Zbl 0292.26011 [2] Podlubny, I., Fractional Differential Equations, (1999), Academic Press · Zbl 0918.34010 [3] Herrmann, R., Fractional Calculus, (2011), World Scientific [4] Hilfer, R., Fractional Calculus, Applications in Physics, (2000), World Scientific · Zbl 0998.26002 [5] Laskin, N., Phys. Rev. E, 62, 3135, (2000) [6] Laskin, N., Chaos, 10, 780, (2000) · Zbl 1071.81513 [7] Laskin, N., Phys. Rev. E, 66, 056108, (2002) [8] Laskin, N., (2010) [9] Naber, M., J. Math. Phys., 45, 3339, (2004) · Zbl 1071.81035 [10] Bayin, S. S., J. Math. Phys., 53, 042105, (2012) · Zbl 1275.81026 [11] Bayin, S. S., J. Math. Phys., 53, 084101, (2012) · Zbl 1278.81056 [12] Bayin, S. S., J. Math. Phys., 54, 074101, (2013) · Zbl 1284.81104 [13] Bayin, S. S., J. Math. Phys., 54, 092101, (2013) · Zbl 1284.81105 [14] Jeng, M.; Xu, S.-L.-Y.; Hawkins, E.; Schwarz, J. M., J. Math. Phys., 51, 062102, (2010) · Zbl 1311.81114 [15] Hawkins, E.; Schwarz, J. M., J. Math. Phys., 54, 014101, (2013) · Zbl 1280.81040 [16] Wei, Y., Phys. Rev. E, 93, 066103, (2016) [17] Wei, Y., Int. J. Theor. Math. Phys., 5, 5, 87, (2015) [18] Dong, J., (2013) [19] Zaba, M.; Garbaczewski, P., J. Math. Phys., 56, 123502, (2015) · Zbl 1332.34133 [20] Luchko, Y., J. Math. Phys., 54, 012111, (2013) · Zbl 1280.81044 [21] Mainardi, F.; Luchko, Yu.; Pagnini, P., Frac. Calculus Appl. Anal., 4, 153, (2001) [22] Herrmann, R., Gam. Ori. Cron. Phys., 1, 1, (2013) [23] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers, New York, London · Zbl 0818.26003 [24] Gorenflo, R. and Mainardi, F., Essentials of Fractional Calculus, 2000, . · Zbl 1030.26004 [25] Gorenflo, R.; Mainardi, F.; Carpinteri, A.; Mainardi, F., Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics, 223-276, (1997), Springer Verlag, Wien [26] Bayin, S. S., Mathematical Methods in Science and Engineering, (2006), Wiley · Zbl 1180.00002 [27] Iomin, A., Chaos, Solitons Fractals, 71, 73, (2015) · Zbl 1352.60067 [28] Bjorken, J. D.; Drell, S. D., Relativistic Quantum Mechanics, 4, (1964), McGraw-Hill [29] Merzbacher, E., Quantum Mechanics, 567, (1970), Wiley
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.