On the location of spectral edges in Z-periodic media
Pavel Exner
Peter Kuchment
Brian Winn
2134/15454
https://repository.lboro.ac.uk/articles/journal_contribution/On_the_location_of_spectral_edges_in_Z-periodic_media/9386465
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.
2014-08-06 11:52:52
untagged
Mathematical Sciences not elsewhere classified