×

Fourier transforms, fractional derivatives, and a little bit of quantum mechanics. (English) Zbl 1479.46053

Summary: We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions \(\mathcal{\mathcal{S}} (\mathbb{R})\), and then we extend it to its dual set, \(\mathcal{\mathcal{S}}'(\mathbb{R})\), the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context.

MSC:

46F10 Operations with distributions and generalized functions
26A33 Fractional derivatives and integrals
46N50 Applications of functional analysis in quantum physics
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] R. A. Adams and J. J. F. Fournier, Sobolev spaces, vol. 140, Second ed., Pure and Applied Mathematics, Elsevier, 2003.
[2] F. Bagarello, “Deformed canonical (anti-)commutation relations and non-self-adjoint Hamiltonians”, pp. 121-188 in Non-selfadjoint operators in quantum physics, Wiley, Hoboken, NJ, 2015. · Zbl 1329.81021
[3] F. Bagarello, “\(kq\)-representation for pseudo-bosons, and completeness of bi-coherent states”, J. Math. Anal. Appl. 450:1 (2017), 631-646. · Zbl 1358.81172
[4] F. Bagarello, F. Gargano, S. Spagnolo, and S. Triolo, “Coordinate representation for non-Hermitian position and momentum operators”, Proc. A. 473:2205 (2017), 20170434, 13. · Zbl 1402.81040
[5] R. A. Brewster and J. D. Franson, “Generalized delta functions and their use in quantum optics”, J. Math. Phys. 59:1 (2018), 012102, 17. · Zbl 1380.81506
[6] R. Herrmann, Fractional calculus: an introduction for physicists, World Scientific, Hackensack, NJ, 2011. · Zbl 1232.26006
[7] R. Hilfer (editor), Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
[8] N. Laskin, “Fractional quantum mechanics”, Phys. Rev. E 62:3 (2000), 3135-3145.
[9] N. Laskin, “Fractional Schrödinger equation”, Phys. Rev. E (3) 66:5 (2002), art.,id.,056108.
[10] N. Laskin, Fractional quantum mechanics, World Scientific, Hackensack, NJ, 2018. · Zbl 1425.81007
[11] I. V. Lindell, “Delta function expansions, complex delta functions and the steepest descent method”, Am. J. Phys. 61 (1993), 438-442.
[12] A. Matos-Abiage, “Fractional dimensional momentum operator for a system of one degree of freedom”, Phys. Scripta 62 (2000), 106-107.
[13] V. Namias, “The fractional order Fourier transform and its application to quantum mechanics”, J. Inst. Math. Appl. 25:3 (1980), 241-265. · Zbl 0434.42014
[14] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications, Edited and with a foreword by S. M. Nikol’skiĭ, Translated from the 1987 Russian original, Revised by the authors. · Zbl 0818.26003
[15] V. A. Smagin, “Complex delta function and its information application”, Autom. Control Comput. Sci. 48:1 (2014), 10-16.
[16] C.-C. Tseng, S.-C. Pei, and S.-C. Hsia, “Computation of fractional derivatives using Fourier transform and digital FIR differentiator”, Signal Processing 80:1 (2000), 151-159. · Zbl 1037.94524
[17] P. Vigué, “Fourier transform of Weyl fractional derivatives”, 2018, https://hal.archives-ouvertes.fr/hal-01740424.
[18] Y. Wei, “The infinite square well problem in the standard, fractional, and relativistic quantum mechanics”, Int. J. Theoretical Math. Phys. 5:4 (2015), 58-65.
[19] Y.
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.