Theoretical and Mathematical Physics, 192(1): 1028–1038 (2017)
REANALYSIS OF AN OPEN PROBLEM ASSOCIATED WITH THE
and K. Pichaghchi
It was recently shown that there are some diﬃculties in the solution method proposed by Laskin for ob-
taining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schr¨odinger
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 Schr¨odinger equation, inﬁnite potential well, Riesz fractional derivative, Mittag–
Feynman and Hibbs  reconstructed the Schr¨odinger equation using an approach based on the path
integral and considering a Gaussian probability distribution. This approach was subsequently extended
by Laskin ,  in the formulation of the fractional Schr¨odinger equation by generalizing the Feynman
path integrals from Brownian-like to L´evy-like quantum mechanical paths. The Schr¨odinger equation thus
obtained contains fractional derivatives with respect to space and time.
In ,  as in many other papers by diﬀerent authors, the fractional Schr¨odinger equation with
fractional derivatives with respect to space and also with respect to space and time was studied with the
quantum Riesz derivative and some speciﬁc potential ﬁelds including a zero potential (free particle), a δ-
potential, an inﬁnite potential well, a Coulomb potential, and a rectangular barrier. Jeng et al.  recently
argued that solutions of the space-fractional Schr¨odinger equation obtained in a piecewise-continuous fashion
are incorrect. Their arguments were based on contradictions arising when the ground state wave function
of the problem with an inﬁnite potential well is substituted in the space-fractional Schr¨odinger equation.
They concluded that many exact solutions in the literature are wrong with the exception of the solution
corresponding to a δ-function potential. This not only casts doubt on existing solutions in the literature
but also makes it very diﬃcult to ﬁnd a meaningful solution. In particular, Luchko  showed that the
Faculty of Mathematical Sciences, University of Malayer, Malayer, Iran, e-mail: email@example.com,
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i
Matematicheskaya Fizika, Vol. 192, No. 1, pp. 103–114, July, 2017. Original article submitted May 11, 2016.
2017 Pleiades Publishing, Ltd.