Estimates of bands for Laplacians on periodic equilateral metric graphs

  • Korotyaev E
  • Saburova N
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Abstract

We consider Laplacians on periodic equilateral metric graphs. The spectrum of the Laplacian consists of an absolutely continuous part (which is a union of an infinite number of non-degenerated spectral bands) plus an infinite number of flat bands, i.e., eigenvalues of infinite multiplicity. We estimate the Lebesgue measure of the bands on a finite interval in terms of geometric parameters of the graph. The proof is based on spectral properties of discrete Laplacians.

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Korotyaev, E., & Saburova, N. (2015). Estimates of bands for Laplacians on periodic equilateral metric graphs. Proceedings of the American Mathematical Society, 144(4), 1605–1617. https://doi.org/10.1090/proc/12815

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