Spectrum of periodic elliptic operators with distant perturbations in space

Golovina, A.

doi:10.1090/S1061-0022-2014-01314-3

Spectrum of periodic elliptic operators with distant perturbations in space
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by A. M. Golovina
Translated by: S. Kislyakov

St. Petersburg Math. J. 25 (2014), 735-754

DOI: https://doi.org/10.1090/S1061-0022-2014-01314-3

Published electronically: July 18, 2014

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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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