Geodesic
Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters
Sbornik. Mathematics, Tome 49 (1984) no. 1, pp. 61-72 
D. G. Vasil'ev. Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters. Sbornik. Mathematics, Tome 49 (1984) no. 1, pp. 61-72. http://geodesic.mathdoc.fr/item/SM_1984_49_1_a4/
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     title = {Asymptotic behavior of the spectrum of pseudodifferential operators with small parameters},
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     year = {1984},
     volume = {49},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1984_49_1_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $A_k$, $k=0,1,\dots,l$, 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 $\varepsilon$, $h_j$, $j=1,2,\dots,l$, 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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[2] Hörmander L., “The spectral function of an elliptic operator”, Acta Math., 121:3–4 (1968), 193–218 ; Khërmander L., “Spektralnaya funktsiya ellipticheskogo operatora”, Matematika, 13:6 (1969), 114–137 | DOI | MR | Zbl

[3] Vasilev D. G., “Asimptotika funktsii raspredeleniya spektra psevdodifferentsialnykh operatorov s parametrami”, Funkts. analiz, 14:3 (1980), 65–66 | MR

[4] Aslanyan A. G., Vasilev D. G., Lidskii V. B., “Chastoty svobodnykh kolebanii tonkoi obolochki, vzaimodeistvuyuschei s zhidkostyu”, Funkts. analiz, 15:3 (1981), 1–9 | MR | Zbl

[5] Aslanyan A. G., Vasilev D. G., Lidskii V. B., “Kolebaniya tonkikh obolochek, vzaimodeistvuyuschikh s zhidkostyu”, UMN, 35:2 (212) (1980), 255–256 | MR

[6] Aslanyan A. G., Vasilev D. G., Lidskii V. B., “Chastoty svobodnykh kolebanii obolochek, vzaimodeistvuyuschikh s zhidkostyu”, Trudy XII Vsesoyuznoi konferentsii po teorii obolochek i plastin, t. 1, EGU, Erevan, 1980, 86–92

[7] Glazman I. M., Pryamye metody kachestvennogo spektralnogo analiza singulyarnykh differentsialnykh operatorov, Fizmatgiz, M., 1963 | MR

[8] Duistermaat J. J., Guillemin V. W., “The spectrum of positive elliptic operators and periodic bicharacteristics”, Invent. Math., 29:1 (1975), 39–79 | DOI | MR | Zbl

[9] Aslanyan A. G., Vasilev D. G., Lidskii V. B., “Raspredelenie sobstvennykh chastot tonkoi uprugoi obolochki, vzaimodeistvuyuschei s zhidkostyu”, UMN, 36:4 (220) (1981), 216

[10] Vasilev D. G., Lidskii V. B., “Kolebaniya tonkikh obolochek, kontaktiruyuschikh s zhidkostyu”, UMN, 37:5 (227) (1982), 224–225