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On the location of spectral edges in Z-periodic media
Periodic second-order ordinary differential operators on R are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multi-dimensional analog of this property is false, as was shown in a 2007 paper by some of the authors of th...
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| Main Authors: | , , |
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| Format: | Default Article |
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2010
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| Online Access: | https://hdl.handle.net/2134/15454 |
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| _version_ | 1818173415535673344 |
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| author | Pavel Exner Peter Kuchment Brian Winn |
| author_facet | Pavel Exner Peter Kuchment Brian Winn |
| author_sort | Pavel Exner (7161647) |
| collection | Figshare |
| description | Periodic second-order ordinary differential operators on R are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multi-dimensional analog of this property is false, as was shown in a 2007 paper by some of the authors of this paper. However, one sometimes encounters the claims that in the case of a single periodicity (i.e., with respect to the lattice Z), the 1D property still holds, and spectral edges occur at the periodic and anti-periodic spectra only. In this work, we show that even in the simplest case of quantum graphs this is not true. It is shown that this is true if the graph consists of a 1D chain of finite graphs connected by single edges, while if the connections are formed by at least two edges, the spectral edges can already occur away from the periodic and anti-periodic spectra. |
| format | Default Article |
| id | rr-article-9386465 |
| institution | Loughborough University |
| publishDate | 2010 |
| record_format | Figshare |
| spelling | rr-article-93864652010-01-01T00:00:00Z On the location of spectral edges in Z-periodic media Pavel Exner (7161647) Peter Kuchment (7161650) Brian Winn (1247334) Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified Periodic second-order ordinary differential operators on R are known to have the edges of their spectra to occur only at the spectra of periodic and antiperiodic boundary value problems. The multi-dimensional analog of this property is false, as was shown in a 2007 paper by some of the authors of this paper. However, one sometimes encounters the claims that in the case of a single periodicity (i.e., with respect to the lattice Z), the 1D property still holds, and spectral edges occur at the periodic and anti-periodic spectra only. In this work, we show that even in the simplest case of quantum graphs this is not true. It is shown that this is true if the graph consists of a 1D chain of finite graphs connected by single edges, while if the connections are formed by at least two edges, the spectral edges can already occur away from the periodic and anti-periodic spectra. 2010-01-01T00:00:00Z Text Journal contribution 2134/15454 https://figshare.com/articles/journal_contribution/On_the_location_of_spectral_edges_in_Z-periodic_media/9386465 CC BY-NC-ND 4.0 |
| spellingShingle | Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified Pavel Exner Peter Kuchment Brian Winn On the location of spectral edges in Z-periodic media |
| title | On the location of spectral edges in Z-periodic media |
| title_full | On the location of spectral edges in Z-periodic media |
| title_fullStr | On the location of spectral edges in Z-periodic media |
| title_full_unstemmed | On the location of spectral edges in Z-periodic media |
| title_short | On the location of spectral edges in Z-periodic media |
| title_sort | on the location of spectral edges in z-periodic media |
| topic | Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified |
| url | https://hdl.handle.net/2134/15454 |