Fractional Schödinger equation for heterogeneous media and Lévy like distributions

We investigate an extension of the Schrödinger equation by considering a fractional differential operator for the spatial variable, which simultaneously takes the heterogeneity of the media and Lévy like distributions into account. By using the Green's function method, we obtain solutions to the equation in the case of the free particle and when it is subject to a delta potential. We also consider a non-local contribution added to the delta potential to explore its influence on the wave function. The solutions show a rich class of spreading behaviors, considerably different from the usual wave function, which may be connected with power-laws and stretched exponential distributions.