Abstract
The variational properties of the spectra of a class of quadratic pencils are investigated. These operator pencils are not strongly damped, which is expressed in a considerable manner in its properties. The obtained results are fundamental in the investigation of two-parameter pencils of waveguide type, which model pencils arising in the theory of regular waveguides. The considerable difficulties, arising at the investigation of pencils of waveguide type, are explained by the fact that they do not generate Rayleigh systems in the entire space, but only on certain of its nonconvex homogeneous sets. These sets occur here as the sets of the admissible vectors of the corresponding extremal problems.
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References
Yu. Sh. Abramov, “Two-parameter operator pencils of waveguide type,” Dokl. Akad. Nauk SSSR,286, No. 4, 777–781 (1986).
A. S. Zil'bergleit and Yu. I. Kopilevich, The Spectral Theory of Regular Waveguides [in Russian], Akad. Nauk SSSR, Fiz.-Tekh. Inst., Leningrad (1983).
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Additional information
Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 80–96, 1990.
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Abramov, Y.S. Pencils of waveguide type and related extremal problems. J Math Sci 64, 1278–1288 (1993). https://doi.org/10.1007/BF01098020
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DOI: https://doi.org/10.1007/BF01098020