2014/06/29 by Evgeny Korotyaev, Korotyaev, Evgeny, Natalia Saburova +1
Mathematics · #FOS: Mathematics #Graph theory and applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.SP
paper · pdf · doi:10.48550/arxiv.1406.7523
18 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1310.3461, arXiv:1312.6510
arxiv created 2014/06/29 · openalex publication_date 2014/06/29 · arxiv updated 2014/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider Laplacians on periodic metric graphs with unit-length edges. The spectrum of these operators 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. Our main result is a localization of spectral bands in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph. The proof is based on the spectral band localization for discrete Laplacians and on the relation between the spectra of discrete and metric Laplacians.