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.
|Original language||English (US)|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|State||Published - Nov 9 2010|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-CI-016-04
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.