Abstract
In this paper, we deal with a fractional Schrödinger equation that contains the quantum Riesz-Feller derivative instead of the Laplace operator in the case of a particle moving in a potential field. In particular, this equation is solved for a free particle in terms of the Fox H-function. On the other hand, we show that from physical viewpoint, the fractional Schrödinger equation with the quantum Riesz-Feller derivative of order α, 0 < α ≤ 2 and skewness θ makes sense only if it reduces to the Laplace operator (α = 2) or to the quantum Riesz fractional derivative (θ = 0). The reason is that the quantum Riesz-Feller derivative is a Hermitian operator and possesses real eigenvalues only when α = 2 or θ = 0. We then focus on the time-independent one-dimensional fractional Schrödinger equation with the quantum Riesz derivative in the case of a particle moving in an infinite potential well. In particular, we show that the explicit formulas for the eigenvalues and eigenfunctions of the time-independent fractional Schrödinger equation that some authors recently claimed to receive cannot be valid. The problem to find right formulas is still open.
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Al-Saqabi, B., Boyadjiev, L. & Luchko, Y. Comments on employing the Riesz-Feller derivative in the Schrödinger equation. Eur. Phys. J. Spec. Top. 222, 1779–1794 (2013). https://doi.org/10.1140/epjst/e2013-01963-3
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DOI: https://doi.org/10.1140/epjst/e2013-01963-3