Spectrum of a Dilated Honeycomb Network
We analyze spectrum of Laplacian supported by a periodic honeycomb lattice with generally unequal edge lengths and a δ 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.
Mathematics Subject ClassificationPrimary 81Q35 Secondary 34B45 34K13 35B10
KeywordsQuantum graphs Hexagon lattice Laplace operator Vertex δ-coupling Spectrum
Unable to display preview. Download preview PDF.
- 2.Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Amer. Math. Soc., Providence (2013)Google Scholar
- 5.Kuchment P., private communication following the publication of Google Scholar
- 10.Schmidt W.M.: Diophantine Approximations and Diophantine Equations Lecture Notes in Mathematics, vol. 1467. Springer, Berlin (1991)Google Scholar
- 11.de Verdière Y.C.: Semi-classical measures on Quantum Graphs and the Gauss map of the determinant manifold (preprint). arXiv:1311.5449