Loughborough University
Browse
preprint.pdf (108.78 kB)

On the location of spectral edges in Z-periodic media

Download (108.78 kB)
journal contribution
posted on 2014-08-06, 11:52 authored by Pavel Exner, Peter Kuchment, Brian WinnBrian Winn
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.

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 THEORETICAL

Volume

43

Issue

47

Pages

? - ? (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 Publishing

Version

  • AM (Accepted Manuscript)

Publication date

2010

ISSN

1751-8113

Language

  • en

Usage metrics

    Loughborough Publications

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC