Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: math
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn