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 Grün...
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| Published in: | Physical review. E, Statistical, nonlinear, and soft matter physics Vol. 88; no. 1; p. 012120 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
United States
01-07-2013
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| Online Access: | Get full text |
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| Summary: | 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. |
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| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 1539-3755 1550-2376 |
| DOI: | 10.1103/PhysRevE.88.012120 |