Fractional schrödinger equation with zero and linear potentials

Abstract

This paper is about the fractional Schrödinger equation expressed in terms of the Caputo time-fractional and quantum Riesz-Feller space fractional derivatives for particle moving in a potential field. The cases of free particle (zero potential) and a linear potential are considered. For free particle, the solution is obtained in terms of the Fox H-function. For the case of a linear potential, the separation of variables method allows the fractional Schrödinger equation to be split into space fractional and time fractional ones. By using the Fourier and Mellin transforms for the space equation and the contour integrals technique for the time equation, the solutions are obtained also in terms of the Fox H-function. Moreover, some recent results related to the time fractional equation have been revised and reconsidered. The results obtained in this paper contain as particular cases already known results for the fractional Schrödinger equation in terms of the quantum Riesz space-fractional derivative and standard Laplace operator.

Two‐sided and regularised Riesz‐Feller derivatives

“…Proof 2.1. In fact, from ( 15), (18), and ( 19), we conclude that the Riesz and Feller derivatives form a base…”

“…A similar procedure has been used in quantum mechanics for studying the fractional Schrödinger equation. 13,14,[16][17][18][19][20][29][30][31] In these studies, the fractional space derivative was implemented through the fractional laplacian, expressed in the frequency domain, by Ψ ( ) and called fractional quantum derivative.…”

“…33 Remark 2.1. Relations ( 15), (18), and ( 19) lead us to deduce the interesting result Corollary 2.1. If > −1, ∈ R, and n ∈ Z, then…”

“…The Riesz-Feller potentials are very useful in several physical applications as it is the case of diffusion [8][9][10][11][12] and quantum mechanics. [13][14][15][16][17][18][19][20] In these fields, partial differential equations are used to describe the underlying phenomena by using Riemann-Liouville (RL) and mainly Caputo (C) derivatives for the time derivative and the Riesz-Feller potential for space derivative. In the general fractional time-space derivatives case, the solution is expresed in terms of Mellin-Barnes integrals.…”

Factorization of the Riesz-Feller fractional quantum harmonic oscillators

“…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. Since then, several authors have dealt with the spatial fractional Schrödinger equation with different types of fractional derivatives and various potentials presenting contradictory results and arguments [2,3,4,5,6,7].…”