High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph

Details High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph

2010-06-08

arxiv.org/abs/1006.1446v2

Physics

Mathematical Physics

28 pages, 1 figure; minor improvements, to appear in J. Phys. A: Math. Theor

Scientific paper

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 $\mathcal{O}(n^j),\, j=1,0,-1,-2,-3$, as the band index $n\to\infty$. The gaps may be of asymptotically constant width or linearly growing with the latter case being generic.

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