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 Mat. Zametki, 2009, Volume 85, Issue 3, Pages 451–455 (Mi mz4670)

On Spectral Properties of the Discrete Schrödinger Operator with Pure Imaginary Finite Potential

Saint-Petersburg State University

Abstract: In this paper, we consider the spectral properties of the discrete Schrödinger operator in the space of square integrable two-sided sequences with a pure imaginary potential of finite rank with zero mean value. We show that if such potentials are small, then the spectrum of the operator under study coincides with the spectrum of the unperturbed operator, and the operator itself is similar to a self-adjoint operator.

Keywords: discrete Schrödinger operator, spectral problem, $\mathscr{PT}$-symmetric potential, similarity to a self-adjoint operator

DOI: https://doi.org/10.4213/mzm4670

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English version:
Mathematical Notes, 2009, 85:3, 437–440

Bibliographic databases:

UDC: 517.948.35

Citation: M. M. Faddeev, “On Spectral Properties of the Discrete Schrödinger Operator with Pure Imaginary Finite Potential”, Mat. Zametki, 85:3 (2009), 451–455; Math. Notes, 85:3 (2009), 437–440

Citation in format AMSBIB
\Bibitem{Fad09} \by M.~M.~Faddeev \paper On Spectral Properties of the Discrete Schr\"odinger Operator with Pure Imaginary Finite Potential \jour Mat. Zametki \yr 2009 \vol 85 \issue 3 \pages 451--455 \mathnet{http://mi.mathnet.ru/mz4670} \crossref{https://doi.org/10.4213/mzm4670} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2548052} \zmath{https://zbmath.org/?q=an:1177.35056} \elib{http://elibrary.ru/item.asp?id=15308692} \transl \jour Math. Notes \yr 2009 \vol 85 \issue 3 \pages 437--440 \crossref{https://doi.org/10.1134/S0001434609030146} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000266561100014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70049088859} 

• http://mi.mathnet.ru/eng/mz4670
• https://doi.org/10.4213/mzm4670
• http://mi.mathnet.ru/eng/mz/v85/i3/p451

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Yan Zh., “Complex $\mathcal{PT}$-symmetric extensions of the nonlinear ultra-short light pulse model”, J. Phys. A, 45:44 (2012), 444035
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