Fractional Schr\"{o}dinger Equation with Zero and Linear Potentials

This paper is about the fractional Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger equation in terms of the quantum Riesz space-fractional derivative and standard Laplace operator.