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Spectral band localization for Schrödinger operators on discrete periodic graphs

Authors: Evgeny Korotyaev and Natalia Saburova
Journal: Proc. Amer. Math. Soc. 143 (2015), 3951-3967
MSC (2010): Primary 47A10
Published electronically: March 27, 2015
MathSciNet review: 3359585
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Abstract: We consider Schrödinger operators on periodic discrete graphs. It is known that the spectrum of these operators has band structure. We describe a localization of spectral bands and estimate the Lebesgue measure of the spectrum in terms of eigenvalues of Dirichlet and Neumann operators on a fundamental domain of the periodic graph. The proof is based on the Floquet decomposition of Schrödinger operators and the minimax principle.

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Additional Information

Evgeny Korotyaev
Affiliation: Department of Mathematical Physics, Faculty of Physics, St. Petersburg State University, Ulianovskaya 2, St. Petersburg, 198904, Russia

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

Keywords: Schr\"odinger operator, periodic discrete graph, spectral band localization
Received by editor(s): October 13, 2013
Received by editor(s) in revised form: May 18, 2014
Published electronically: March 27, 2015
Communicated by: Joachim Krieger
Article copyright: © Copyright 2015 American Mathematical Society

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