Comment on “On the consistency of the solutions of the space fractional Schrödinger equation” [J. Math. Phys. 53, 042105 (2012)]

Abstract

We give additional details about how these integrals are evaluated and show that there is no inconsistency for an infinite square well. C 2013 AIP Publishing LLC.

Fractional Quantum Mechanics

“…which can be presented as geometric series (the range of convergence is given by the criteria |Es −β /i βh β | < 1, see, [12]),…”

Definition of the Riesz derivative and its application to space fractional quantum mechanics

“…The last step naturally alters the analytic structure of the Riesz derivative and as in the box problem leads to misleading results [15,16,28]. In such situations, using the original expression of the Riesz derivative [10][11][12][13] that accommodates analytic continuation allows correct implementation of the contour integral theorems, thus resolving the controversy.…”

“…This form of the Riesz derivative allows analytic continuation and thus the correct implementation of the complex contour integral theorems becomes possible [10][11][12][13]27]. For real ω, F {R α x f (x)} [Eq.…”

“…Successful applications to anomalous diffusion and evolution problems [1][2][3][4] were immediately followed by applications to quantum mechanics [5][6][7][8][9][10][11][12]. In particular, Laskin's space fractional quantum mechanics is intriguing since it also follows from Feynman's path integral approach over Lévy paths [5].…”

“…One of the crucial elements of the space fractional quantum mechanics is the Riesz derivative. We have shown that a particular representation of the Riesz derivative that accommodates analytic continuation can be used to evaluate the integral in question, thus resolving the so-called inconsistency problem [10][11][12][13]. Misleading conclusions regarding the Laskin's space fractional quantum mechanics [5] often results when one ignores the basic assumptions and restrictions involved in the use of the Riesz derivative [15][16][17][18][19][20][21].…”