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Estimates of bands for Laplacians on periodic equilateral metric graphs


Authors: Evgeny Korotyaev and Natalia Saburova
Journal: Proc. Amer. Math. Soc. 144 (2016), 1605-1617
MSC (2010): Primary 47A10
DOI: https://doi.org/10.1090/proc/12815
Published electronically: August 12, 2015
MathSciNet review: 3451237
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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-degenerate 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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Additional Information

Evgeny Korotyaev
Affiliation: Mathematical Physics Department, Faculty of Physics, Ulianovskaya 2, St. Petersburg State University, St. Petersburg, 198904, Russia
Email: korotyaev@gmail.com

Natalia Saburova
Affiliation: Department of Mathematical Analysis, Algebra and Geometry, Institute of Mathematics, Information and Space Technologies, Uritskogo St. 68, Northern (Arctic) Federal University, Arkhangelsk, 163002, Russia
Email: n.saburova@gmail.com

DOI: https://doi.org/10.1090/proc/12815
Keywords: Spectral bands, flat bands, Laplace operator, periodic equilateral metric graph
Received by editor(s): February 22, 2015
Received by editor(s) in revised form: April 23, 2015
Published electronically: August 12, 2015
Additional Notes: This research was supported by the RSF grant no. 15-11-30007
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

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