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Degenerate band edges in periodic quantum graphs

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Abstract

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.

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Notes

  1. These features are also called “threshold effects” [13] whenever they depend only on the infinitesimal structure (e.g., a finite number of Taylor coefficients) of the dispersion relation at the spectral edges.

  2. This is a consequence of Barra–Gaspard ergodicity of quantum graphs: informally, what happens once for one choice of lengths will happen with finite frequency for almost all choices of lengths. For more precise statements, see [3, 5, 11, 16].

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Acknowledgements

The work of the first author was partially supported by NSF DMS–1815075 Grant, and the work of the second author was partially supported by the AMS-Simons Travel Grant. Both authors express their gratitude to Peter Kuchment for introducing them to this exciting topic and to Lior Alon and Ram Band for many deep discussions. We thank an anonymous referee for several improving suggestions.

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  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Gregory Berkolaiko

  2. Department of Mathematics, The University of Arizona, Tucson, AZ, 85721-0089, USA

    Minh Kha

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  1. Gregory Berkolaiko
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  2. Minh Kha
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Correspondence to Minh Kha.

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Berkolaiko, G., Kha, M. Degenerate band edges in periodic quantum graphs. Lett Math Phys 110, 2965–2982 (2020). https://doi.org/10.1007/s11005-020-01312-x

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