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