Some solutions to the space fractional Schrödinger equation using momentum representation method

The space fractional Schrödinger equation with linear potential, delta-function potential, and Coulomb potential is studied under momentum representation using Fourier transformation. By use of Mellin transform and its inverse transform, we obtain the energy levels and wave functions expressed in $H$ function for a particle in linear potential field. The wave function expressed also by the $H$ function and the unique energy level of the bound state for the particle of even parity state in delta-function potential well, which is proved to have no action on the particle of odd parity state, is also obtained. The integral form of the wave functions for a particle in Coulomb potential field is shown and the corresponding energy levels which have been discussed in Laskin’s paper [Phys. Rev. E 66, 056108 (2002)] are proved to satisfy an equality of infinite limit of the $H$ function. All of these results contain the ones of the standard quantum mechanics as their special cases.