Some properties of the one-dimensional L\'{e}vy crystal

We introduce and discuss the one-dimensional L\'{e}vy crystal as a probable candidate for an experimentally accessible realization of space fractional quantum mechanics (SFQM) in a condensed matter environment. The discretization of the space fractional Schr\"{o}dinger equation with the help of shifted Gr\"{u}nwald-Letnikov derivatives delivers a straight-forward route to define the L\'{e}vy crystal of order $\alpha \in (1,2]$. As key ingredients for its experimental identification we study the dispersion relation as well as the density of states for arbitrary $\alpha \in (1,2]$. It is demonstrated that in the limit of small wavenumbers all interesting properties of continuous space SFQM are recovered, while for $\alpha \to 2$ the well-established nearest neighbor one-dimensional tight binding chain arises.