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

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