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

Pavel Exner, Peter Kuchment, Brian Winn

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)474022
JournalJournal of Physics A: Mathematical and Theoretical
Volume43
Issue number47
DOIs
StatePublished - Nov 9 2010
Externally publishedYes

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