 1086 Pavel Exner, Peter Kuchment, Brian Winn
 On the location of spectral edges in $\mathbb{Z}$periodic media
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Jun 11, 10

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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
multidimensional 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
antiperiodic 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 antiperiodic spectra.
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