On the location of spectral edges in Z-periodic media

0350853 - ÚJF 2011 RIV GB eng J - Journal Article
Exner, Pavel - Kuchment, P. - Winn, B.
On the location of spectral edges in Z-periodic media.
Journal of Physics A-Mathematical and Theoretical. Roč. 43, č. 47 (2010), 474022/1-474022/8. ISSN 1751-8113. E-ISSN 1751-8121
R&D Projects: GA MŠMT LC06002
Institutional research plan: CEZ:AV0Z10480505
Keywords : OPERATORS * GRAPHS
Subject RIV: BE - Theoretical Physics
Impact factor: 1.641, year: 2010

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.
Permanent Link: http://hdl.handle.net/11104/0190742