Abstract
In a space of vector functions, we consider the spectral problem
, where
, and the a αjk and p jk are constants, x ∈ Ω, and Ω is a bounded open set. The boundary conditions correspond to the Dirichlet problem. Let N ±(μ) be the positive and negative spectral counting functions. We establish the asymptotics N ±(μ) ∼ (mesmΩ)φ±(μ) as μ → +0. The functions φ±(μ) are independent of Ω. In the nonelliptic case, these asymptotics are in general different from the classical (Weyl) asymptotics.
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References
M. Sh. Birman and M. Z. Solomyak, in: Itogi Nauki i Tekhniki. Math. Analysis [in Russian], vol. 14, VINITI, Moscow, 1977, 5–58.
A. S. Andreev, Mat. Sb., 197:2 (2006), 17–34; English transl.: Sb. Math., 197:1–2 (2006), 153–171.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 2, pp. 75–78, 2008
Original Russian Text Copyright © by A. S. Andreev
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Andreev, A.S. Quasi-Weyl asymptotics of the spectrum of the vector Dirichlet problem. Funct Anal Its Appl 42, 141–143 (2008). https://doi.org/10.1007/s10688-008-0020-8
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DOI: https://doi.org/10.1007/s10688-008-0020-8