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

TL;DR

This paper investigates the location of spectral edges in one-dimensional periodic media, demonstrating that unlike in simple cases, spectral edges can occur away from periodic and anti-periodic spectra in quantum graphs with complex connections.

Contribution

It shows that spectral edges in 1D periodic quantum graphs can occur away from the periodic and anti-periodic spectra when the graph connections are more complex.

Findings

Spectral edges can occur away from periodic and anti-periodic spectra in complex quantum graphs.

In simple chain graphs, spectral edges align with periodic and anti-periodic spectra.

Complex connections in quantum graphs lead to different spectral edge locations.

Abstract

Periodic $2$nd 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 article. However, one sometimes encounters the claims that in the case of a single periodicity (i.e., with respect to the lattice $\mathbb{Z}$), 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.