Potential condensed-matter realization of space-fractional quantum mechanics: 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 straightforward route to define the Lévy crystal of order αε(1,2]. As key ingredients for its experimental identification we study the dispersion relation as well as the density of states for arbitrary αε(1,2]. It is demonstrated that in the limit of small wave numbers all interesting properties of continuous-space SFQM are recovered, while for α→2 the well-established nearest-neighbor one-dimensional tight-binding chain arises.