Geodesic
Reanalysis of an open problem associated with the fractional Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 1, pp. 103-114 Cet article a éte moissonné depuis la source Math-Net.Ru

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It was recently shown that there are some difficulties in the solution method proposed by Laskin for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schrödinger equation with an infinite potential well encountered in quantum mechanics. In fact, this problem is still open. We propose a new fractional approach that allows overcoming the limitations of some previously introduced strategies. In deriving the solution, we use a method based on the eigenfunction of the Weyl fractional derivative. We obtain a solution suitable for computations in a closed form in terms of Mittag–Leffler functions and fractional trigonometric functions. It is a simple extension of the results previously obtained by Laskin et al.
Keywords: fractional Schrödinger equation, infinite potential well, Riesz fractional derivative, Mittag–Leffler function. 
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K. Sayevand; K. Pichaghchi. Reanalysis of an open problem associated with the fractional Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 1, pp. 103-114. http://geodesic.mathdoc.fr/item/TMF_2017_192_1_a6/

[1] R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965 | Zbl

[2] N. Laskin, “Fractional quantum mechanics and Lévy path integrals”, Phys. Lett. A, 268:4–6 (2000), 298–305 | DOI | MR | Zbl

[3] N. Laskin, “Fractional Schrödinger equation”, Phys. Rev. E, 66:5 (2002), 056108, 7 pp. | DOI | MR

[4] S. Wang, M. Xu, “Generalized fractional Schrödinger equation with space-time fractional derivatives”, J. Math. Phys., 48:4 (2007), 043502, 10 pp. | DOI | MR | Zbl

[5] J. Dong, M. Xu, “Space-time fractional Schrödinger equation with time-independent potentials”, J. Math. Anal. Appl., 344:2 (2008), 1005–1017 | DOI | MR | Zbl

[6] M. Jeng, S. L. Y. Xu, E. Hawkins, J. M. Schwarz, “On the nonlocality of the fractional Schrödinger equation”, J. Math. Phys., 51:6 (2010), 062102, 6 pp. | DOI | MR | Zbl

[7] Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well”, J. Math. Phys., 54:1 (2013), 012111, 10 pp. | DOI | MR | Zbl

[8] S. S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation”, J. Math. Phys., 53:4 (2012), 042105, 9 pp. | DOI | MR | Zbl

[9] J. T. Machado, F. Mainardi, V. Kiryakova, “Fractional calculus: quo vadimus? (Where are we going?)”, Fract. Calc. Appl. Anal., 18:2 (2015), 495–526 | DOI | MR | Zbl

[10] M. Kwaśnicki, “Eigenvalues of the fractional Laplace operator in the interval”, J. Funct. Anal., 262:5 (2012), 2379–2402 | DOI | MR | Zbl

[11] A. Zoia, A. Rosso, M. Kardar, “Fractional Laplacian in bounded domains”, Phys. Rev. E, 76:2 (2007), 021116 pp. | DOI | MR

[12] R. Herrmann, “The fractional Schrödinger equation and the infinite potential well–numerical results using the Riesz derivative”, Gam. Ori. Chron. Phys., 1 (2013), 1–12, arXiv: 1210.4410

[13] S. Bayin, “On the consistency of the solutions of the space fractional Schrödinger equation”, J. Math. Phys., 53:4 (2012), 042105, 9 pp. | DOI | MR | Zbl

[14] E. Hawkins, J. M. Schwarz, “Comment on 'On the consistency of solutions of the space fractional Schrödinger equation'”, J. Math. Phys., 54:1 (2013), 014101, 5 pp. | DOI | MR | Zbl

[15] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006 | MR

[16] D. Băleanu, O. G. Mustafa, D. O'Regan, “A Kamenev-type oscillation result for a linear (1+$\alpha$)-order fractional differential equation”, Appl. Math. Comput., 259 (2015), 374–378 | DOI | MR

[17] M. Al-Refai, Y. Luchko, “Maximum principle for the multi-term time-fractional diffusion equations with the Riemann–Liouville fractional derivatives”, Appl. Math. Comput., 257 (2015), 40–51 | DOI | MR | Zbl

[18] K. Sayevand, K. Pichaghchi, “Successive approximation: a survey on stable manifold of fractional differential systems”, Fract. Calc. Appl. Anal., 18:3 (2015), 621–641 | DOI | MR | Zbl

[19] X. J. Yang, “Local fractional integral transforms”, Progr. Nonlinear Sci., 4 (2011), 1–225

[20] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publ., Hong Kong, 2011

[21] Ş. S. Bayin, Mathematical Methods in Science and Engineering, John Wiley and Sons, Hoboken, NJ, 2006 | DOI | MR

[22] N. Laskin, “Fractals and quantum mechanics”, Chaos, 10:4 (2000), 780–790 | DOI | MR | Zbl