Some properties of the one-dimensional Lévy crystal

We introduce and discuss the one-dimensional Lévy 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 Schrödinger equation with the help of shifted Grünwald-Letnikov derivatives delivers a straight-forward route to define the Lévy crystal of order $α\in (1,2]$. As key ingredients for its experimental identification we study the dispersion relation as well as the density of states for arbitrary $α\in (1,2]$. It is demonstrated that in the limit of small wavenumbers all interesting properties of continuous space SFQM are recovered, while for $α\to 2$ the well-established nearest neighbor one-dimensional tight binding chain arises.

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