## Schrödinger operators with guided potentials on periodic graphs

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- by Evgeny Korotyaev and Natalia Saburova
- Proc. Amer. Math. Soc.
**145**(2017), 4869-4883 - 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.## References

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## Bibliographic Information

**Evgeny Korotyaev**- Affiliation: Department of Higher Mathematics and Mathematical Physics, Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia
- MR Author ID: 211673
- 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
- MR Author ID: 1073098
- Email: n.saburova@gmail.com; n.saburova@narfu.ru
- Received by editor(s): December 16, 2016
- Published electronically: July 28, 2017
- Communicated by: Joachim Krieger
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**145**(2017), 4869-4883 - MSC (2010): Primary 47A10
- DOI: https://doi.org/10.1090/proc/13733
- MathSciNet review: 3692002