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 Mat. Sb. (N.S.), 1983, Volume 121(163), Number 1(5), Pages 60–71 (Mi msb2154)

Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters

D. G. Vasil'ev

Abstract: The eigenvalue problem
$$L(\varepsilon,h)f\equiv\varepsilon^{m_0}A_0f+\sum^l_{j=1}h_j\varepsilon^{m_j}A_jf=\lambda f.$$
is considered on an $n$-dimensional compact manifold without boundary. Here the $A_k$, $k=0,1,…,l$, are symmetric scalar classical pseudodifferential operators of orders $m_k$ with leading symbols $a_k(x,\xi)$, $m_0>0$, $m_0\geqslant m_k\geqslant0$, $a_0(x,\xi)>0$ and $\varepsilon$, $h_j$, $j=1,2,…,l$, are small real parameters with $\varepsilon>0$ and $h_j=O(\varepsilon^{1/p})$, where $p$ is a positive integer. The distribution functions $n(\lambda,L(\varepsilon,h))$ of the eigenvalues of the operator $L(\varepsilon,h)$ are studied. Let $[\Lambda_1,\Lambda_2]$ be a fixed interval of the positive half-line ($\Lambda_1>0$). An asymptotic formula with optimal relative error $O(\varepsilon)$ is obtained for $n(\lambda,L(\varepsilon,h))$ as $\varepsilon\to0$ when $\lambda\in[\Lambda_1,\Lambda_2]$.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 49:1, 61–72

Bibliographic databases:

UDC: 517.2
MSC: Primary 41A60, 58G15, 58G25; Secondary 35S99, 47G05

Citation: D. G. Vasil'ev, “Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters”, Mat. Sb. (N.S.), 121(163):1(5) (1983), 60–71; Math. USSR-Sb., 49:1 (1984), 61–72

Citation in format AMSBIB
\Bibitem{Vas83}
\by D.~G.~Vasil'ev
\paper Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters
\jour Mat. Sb. (N.S.)
\yr 1983
\vol 121(163)
\issue 1(5)
\pages 60--71
\mathnet{http://mi.mathnet.ru/msb2154}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=699738}
\zmath{https://zbmath.org/?q=an:0559.35060|0534.35075}
\transl
\jour Math. USSR-Sb.
\yr 1984
\vol 49
\issue 1
\pages 61--72
\crossref{https://doi.org/10.1070/SM1984v049n01ABEH002697}

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This publication is cited in the following articles:
1. D. G. Vasil'ev, V. B. Lidskii, “Quasiresonances in the problem of forced vibrations of a thin elastic shell interacting with a liquid”, Funct. Anal. Appl., 20:4 (1986), 267–276
2. Levendorskii S., “The Approximate Spectral Projection Method”, Acta Appl. Math., 7:2 (1986), 137–197
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