Exner P., Turek O.
High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph
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ABSTRACT. We investigate a periodic quantum graph in form of a square
lattice with a general self-adjoint coupling at the vertices. We
analyze the spectrum, in particular, its high-energy behaviour.
Depending on the coupling type, bands and gaps have different
asymptotics. Bands may be flat even if the edges are coupled, and
non-flat band widths may behave as $\O(n^j),\, j=1,0,-1,-2,-3$, as
the band index $n o\infty$. The gaps may be of asymptotically
constant width or linearly growing with the latter case being
generic.