Abstract
Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the non-Hermitian fractional eigenvalue problem in the k space by introducing in that space a new class of Hermite ‘polynomials’ that we call Riesz-Feller Hermite ‘polynomials’. Using the inverse Fourier transform in Mathematica, interesting analytic results for the same eigenvalue problem in the x space are also obtained. Additionally, a more general factorization with two different Lévy indices is briefly introduced.
Highlights
A type of fractional quantum harmonic oscillator has been first discussed by Laskin in one of his breakthrough papers [1] on fractional quantum mechanics, but he tackled only a semiclassical approximation
Olivar-Romero and Rosas-Ortiz [8] were first ones to apply the factorization method [9, 10] to a fractional differential equation choosing precisely the fractional quantum harmonic oscillator as the case study for their considerations
Aαψ0(α)(x) = 0 −→ ddxαα//22 + x ψ0(α)(x) = 0 . (13). Since this is a fractional derivative equation, we will solve it in the k-space by taking into account that the Fourier transform F of the Riesz-Feller derivative dα/dxα of a function is characterized by its specific symbol Ψθα
Summary
A type of fractional quantum harmonic oscillator has been first discussed by Laskin in one of his breakthrough papers [1] on fractional quantum mechanics, but he tackled only a semiclassical approximation. Several authors have dealt with the spatial fractional Schrodinger equation with different types of fractional derivatives and various potentials presenting contradictory results and arguments [2, 3, 4, 5, 6, 7]. In line with Laskin, they used the Riesz fractional derivative reporting some interesting results and making suggestions for future work. This motivated us to proceed with a substantial extension of their results, which we present in this paper.
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