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On the location of spectral edges in Z-periodic media
journal contribution
posted on 2014-08-06, 11:52 authored by Pavel Exner, Peter Kuchment, Brian WinnBrian WinnPeriodic 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.
Funding
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.
History
School
- Science
Department
- Mathematical Sciences
Published in
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICALVolume
43Issue
47Pages
? - ? (8)Citation
EXNER, P., KUCHMENT, P. and WINN, B., 2010. On the location of spectral edges in Z-periodic media. Journal of Physics A - Mathematical & Theoretical, 43, 474022, 8pp.Publisher
© IOP PublishingVersion
- AM (Accepted Manuscript)
Publication date
2010ISSN
1751-8113Publisher version
Language
- en