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

TL;DR

This paper studies the high-energy spectral behavior of a square-lattice quantum graph with various vertex couplings, revealing how band and gap asymptotics depend on coupling types and band indices.

Contribution

It provides a detailed analysis of the asymptotic behavior of bands and gaps in the spectrum of a periodic quantum graph with general self-adjoint vertex couplings.

Findings

Bands can be flat even with edge coupling.

Band widths can scale as (n^j) for j=1,0,-1,-2,-3.

Gaps can be asymptotically constant or grow linearly.

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