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Spectrum of periodic elliptic operators with distant perturbations in space


Author: A. M. Golovina
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 5.
Journal: St. Petersburg Math. J. 25 (2014), 735-754
MSC (2010): Primary 35B20
Published electronically: July 18, 2014
MathSciNet review: 3184606
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Abstract | References | Similar Articles | Additional Information

Abstract: A periodic selfadjoint differential operator of even order and with distant perturbations in a multidimensional space is treated. The role of perturbations is played by arbitrary localized operators. The localization is described by specially chosen weight functions. The behavior of the spectrum of the perturbed operator is studied under the condition that the distance between the domains where the perturbation are localized tends to infinity. It is shown that there exists a simple isolated eigenvalue of the perturbed operator that tends to a simple isolated eigenvalue of the limit operator. Series expansions are obtained for this eigenvalue of the perturbed operator and for the corresponding eigenfunction. Uniform convergence for these series is shown and formulas for their terms are deduced.


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Additional Information

A. M. Golovina
Affiliation: Department of Physics and Mathematics, M. Akmulla Bashkir State Pedagogocal University, ul. Oktyabr$’$skoi revolyutsii 3a, Ufa 450055, Russia
Email: nastya{\textunderscore}gm@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01314-3
Keywords: Selfadjoint operator, distant perturbations, spectrum, eigenvalue, eigenfunction, asymptotics
Received by editor(s): May 19, 2012
Published electronically: July 18, 2014
Additional Notes: Supported by RFBR and by the Federal Thematic Program “Scientific and educational personnel for innovative Russia”, 2009-2013 (contract no. 14.B37.21.0358).
Article copyright: © Copyright 2014 American Mathematical Society