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Schrödinger operators with guided potentials on periodic graphs


Authors: Evgeny Korotyaev and Natalia Saburova
Journal: Proc. Amer. Math. Soc. 145 (2017), 4869-4883
MSC (2010): Primary 47A10
DOI: https://doi.org/10.1090/proc/13733
Published electronically: July 28, 2017
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Abstract: We consider discrete Schrödinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the unperturbed operator is a union of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed operator consists of the ``unperturbed'' one plus the additional guided spectrum, which is a union of a finite number of bands. We estimate the position of the guided bands and their length in terms of graph geometric parameters. We also determine the asymptotics of the guided bands for large guided potentials. Moreover, we show that the possible number of the guided bands, their length and position can be rather arbitrary for some specific potentials.


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Evgeny Korotyaev
Affiliation: Department of Higher Mathematics and Mathematical Physics, Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia
Email: korotyaev@gmail.com; e.korotyaev@spbu.ru

Natalia Saburova
Affiliation: Department of Mathematical Analysis, Algebra and Geometry, Northern (Arctic) Federal University, Severnaya Dvina Emb. 17, Arkhangelsk, 163002, Russia
Email: n.saburova@gmail.com; n.saburova@narfu.ru

DOI: https://doi.org/10.1090/proc/13733
Keywords: Discrete Schr\"odinger operator, periodic graph, guided waves
Received by editor(s): December 16, 2016
Published electronically: July 28, 2017
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 American Mathematical Society

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