On the location of spectral edges in \mathbb {Z}-periodic media

Handle URI:
http://hdl.handle.net/10754/599060
Title:
On the location of spectral edges in \mathbb {Z}-periodic media
Authors:
Exner, Pavel; Kuchment, Peter; Winn, Brian
Abstract:
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 IOP Publishing Ltd.
Citation:
Exner P, Kuchment P, Winn B (2010) On the location of spectral edges in \mathbb {Z}-periodic media. J Phys A: Math Theor 43: 474022. Available: http://dx.doi.org/10.1088/1751-8113/43/47/474022.
Publisher:
IOP Publishing
Journal:
Journal of Physics A: Mathematical and Theoretical
KAUST Grant Number:
KUS-CI-016-04
Issue Date:
9-Nov-2010
DOI:
10.1088/1751-8113/43/47/474022
Type:
Article
ISSN:
1751-8113; 1751-8121
Sponsors:
The authors express their gratitude to G Berkolaiko for his suggestion to discuss the Z-periodic case and for useful comments about the proofs. The work of the first author was supported in part by the Czech Ministry of Education, Youth and Sports within the project LC06002. The work of the second author was partially supported by the KAUST grant KUS-CI-016-04 through the Inst. Appl. Math. Comput. Sci. (IAMCS) at Texas A&M University. The third author has been financially supported by the National Sciences Foundation under research grant DMS-0604859. The authors are grateful to these agencies for the support and the referees for their useful comments.
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorExner, Pavelen
dc.contributor.authorKuchment, Peteren
dc.contributor.authorWinn, Brianen
dc.date.accessioned2016-02-25T13:52:06Zen
dc.date.available2016-02-25T13:52:06Zen
dc.date.issued2010-11-09en
dc.identifier.citationExner P, Kuchment P, Winn B (2010) On the location of spectral edges in \mathbb {Z}-periodic media. J Phys A: Math Theor 43: 474022. Available: http://dx.doi.org/10.1088/1751-8113/43/47/474022.en
dc.identifier.issn1751-8113en
dc.identifier.issn1751-8121en
dc.identifier.doi10.1088/1751-8113/43/47/474022en
dc.identifier.urihttp://hdl.handle.net/10754/599060en
dc.description.abstractPeriodic 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 IOP Publishing Ltd.en
dc.description.sponsorshipThe authors express their gratitude to G Berkolaiko for his suggestion to discuss the Z-periodic case and for useful comments about the proofs. The work of the first author was supported in part by the Czech Ministry of Education, Youth and Sports within the project LC06002. The work of the second author was partially supported by the KAUST grant KUS-CI-016-04 through the Inst. Appl. Math. Comput. Sci. (IAMCS) at Texas A&M University. The third author has been financially supported by the National Sciences Foundation under research grant DMS-0604859. The authors are grateful to these agencies for the support and the referees for their useful comments.en
dc.publisherIOP Publishingen
dc.titleOn the location of spectral edges in \mathbb {Z}-periodic mediaen
dc.typeArticleen
dc.identifier.journalJournal of Physics A: Mathematical and Theoreticalen
dc.contributor.institutionNuclear Physics Institute of the Academy of Sciences of the Czech Republic v. v. i., Rez, Czech Republicen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionLoughborough University, Loughborough,en
kaust.grant.numberKUS-CI-016-04en
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