On the location of spectral edges in \mathbb {Z}-periodic media
Abstract
Periodic second-order ordinary differential operators on R are known to have \nthe edges of their spectra to occur only at the spectra of periodic and antiperiodic \nboundary value problems. The multi-dimensional analog of this \nproperty is false, as was shown in a 2007 paper by some of the authors of \nthis paper. However, one sometimes encounters the claims that in the case of \na single periodicity (i.e., with respect to the lattice Z), the 1D property still \nholds, and spectral edges occur at the periodic and anti-periodic spectra only. \nIn this work, we show that even in the simplest case of quantum graphs this is \nnot true. It is shown that this is true if the graph consists of a 1D chain of finite \ngraphs connected by single edges, while if the connections are formed by at \nleast two edges, the spectral edges can already occur away from the periodic \nand anti-periodic spectra.