Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps
We prove, via an elementary variational method, one-dimensional (1D) and two-dimensional (2D) localization within the band gaps of a periodic Schrödinger operator for any mostly negative or mostly positive defect potential, V, whose depth is not too great compared to the size of the gap. In a simila...
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| Format: | Article |
| Language: | en_US |
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American Physical Society
2010
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| Online Access: | http://hdl.handle.net/1721.1/54686 https://orcid.org/0000-0001-7327-4967 https://orcid.org/0000-0001-5994-0396 |
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| author | Johnson, Steven G. Avniel, Yehuda B. Lee, Karen Ka Yan Parzygnat, Arthur J. |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Johnson, Steven G. Avniel, Yehuda B. Lee, Karen Ka Yan Parzygnat, Arthur J. |
| author_sort | Johnson, Steven G. |
| collection | MIT |
| description | We prove, via an elementary variational method, one-dimensional (1D) and two-dimensional (2D) localization within the band gaps of a periodic Schrödinger operator for any mostly negative or mostly positive defect potential, V, whose depth is not too great compared to the size of the gap. In a similar way, we also prove sufficient conditions for 1D and 2D localization below the ground state of such an operator. Furthermore, we extend our results to 1D and 2D localization in d dimensions; for example, by a linear or planar defect in a three-dimensional crystal. For the case of D-fold degenerate band edges, we also give sufficient conditions for localization of up to D states. |
| first_indexed | 2024-09-23T09:54:03Z |
| format | Article |
| id | mit-1721.1/54686 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:54:03Z |
| publishDate | 2010 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/546862022-09-26T14:25:05Z Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps Johnson, Steven G. Avniel, Yehuda B. Lee, Karen Ka Yan Parzygnat, Arthur J. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Department of Mathematics Massachusetts Institute of Technology. Research Laboratory of Electronics Johnson, Steven G. Johnson, Steven G. Avniel, Yehuda B. Lee, Karen Ka Yan Parzygnat, Arthur We prove, via an elementary variational method, one-dimensional (1D) and two-dimensional (2D) localization within the band gaps of a periodic Schrödinger operator for any mostly negative or mostly positive defect potential, V, whose depth is not too great compared to the size of the gap. In a similar way, we also prove sufficient conditions for 1D and 2D localization below the ground state of such an operator. Furthermore, we extend our results to 1D and 2D localization in d dimensions; for example, by a linear or planar defect in a three-dimensional crystal. For the case of D-fold degenerate band edges, we also give sufficient conditions for localization of up to D states. City University of New York. Macaulay Honors College 2010-04-29T16:50:56Z 2010-04-29T16:50:56Z 2010-04 2010-03 Article http://purl.org/eprint/type/SubmittedJournalArticle 1550-235X 1098-0121 http://hdl.handle.net/1721.1/54686 Parzygnat, Arthur et al. “Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps.” Physical Review B 81.15 (2010): 155324. © 2010 The American Physical Society https://orcid.org/0000-0001-7327-4967 https://orcid.org/0000-0001-5994-0396 en_US http://dx.doi.org/10.1103/PhysRevB.81.155324 Physical Review B Attribution-Noncommercial-Share Alike 3.0 Unported http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf American Physical Society Steven G. Johnson |
| spellingShingle | Johnson, Steven G. Avniel, Yehuda B. Lee, Karen Ka Yan Parzygnat, Arthur J. Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| title | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| title_full | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| title_fullStr | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| title_full_unstemmed | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| title_short | Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| title_sort | sufficient conditions for two dimensional localization by arbitrarily weak defects in periodic potentials with band gaps |
| url | http://hdl.handle.net/1721.1/54686 https://orcid.org/0000-0001-7327-4967 https://orcid.org/0000-0001-5994-0396 |
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