Berkolaiko, Gregory; Kha, Minh Degenerate band edges in periodic quantum graphs. (English) Zbl 1456.81194 Lett. Math. Phys. 110, No. 11, 2965-2982 (2020). Summary: Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet-Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of \({\mathbb{Z}}^3\)-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons. Cited in 1 Document MSC: 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 34B45 Boundary value problems on graphs and networks for ordinary differential equations 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 34L05 General spectral theory of ordinary differential operators 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35P15 Estimates of eigenvalues in context of PDEs 81U30 Dispersion theory, dispersion relations arising in quantum theory Keywords:spectral theory; mathematical physics; quantum graphs; periodic differential operators; maximal abelian coverings; band edges; Floquet-Bloch theory PDFBibTeX XMLCite \textit{G. Berkolaiko} and \textit{M. Kha}, Lett. Math. Phys. 110, No. 11, 2965--2982 (2020; Zbl 1456.81194) Full Text: DOI arXiv References: [1] Ashcroft, NW; Mermin, ND, Solid State Physics (1976), New York: Holt, Rinehart and Winston, New York [2] Baez, J.: Topological crystals. 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