Spectrum of a dilated honeycomb network

Abstract

We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a $\delta$ type coupling in the vertices. Such a quantum graph has nonempty point spectrum with compactly supported eigenfunctions provided all the edge lengths are commensurate. We derive conditions determining the continuous spectral component and show that existence of gaps may depend on number-theoretic properties of edge lengths ratios. The case when two of the three lengths coincide is discussed in detail.

21 pages, one figure

References in corpus (2)

• On the spectra of carbon nano-structures
• Universality of the momentum band density of periodic networks

Cited by in corpus (2)

• Spectral Theory of Infinite Quantum Graphs
• Markov constant and quantum instabilities