This article is cited in 1 scientific paper (total in 1 paper)
Spectrum and completeness of natural oscillations of the atmosphere with temperature stratification
S. A. Stepin
M. V. Lomonosov Moscow State University
Study of the vertical structure of small oscillations of the atmosphere leads to a nonselfadjoint singular boundary eigenvalue problem containing the spectral parameter in the equation and the boundary condition. In this article the problem in question is reduced to the spectral analysis of the operator pencil $\mathcal L(\lambda)=I-\lambda U-(1/\lambda)V$, where $U$ and $V$ are positive compact operators in $L_2(0,\infty)$. By means of a technique based on the study of the energy quadratic form and application of the theory of operator pencils it is proved that the eigenvalues of the problem are real, and their multiplicity is computed; existence of two Riesz bases composed of eigenfunctions is established, and the property of twofold completeness of the system of eigenfunctions and associated functions in the Hilbert space corresponding to the physical formulation of the problem is proved.
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Russian Academy of Sciences. Sbornik. Mathematics, 1994, 79:1, 179–190
MSC: Primary 76C15, 86A10, 34L10; Secondary 47A56
S. A. Stepin, “Spectrum and completeness of natural oscillations of the atmosphere with temperature stratification”, Mat. Sb., 184:6 (1993), 83–98; Russian Acad. Sci. Sb. Math., 79:1 (1994), 179–190
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\paper Spectrum and completeness of natural oscillations of the~atmosphere with temperature stratification
\jour Mat. Sb.
\jour Russian Acad. Sci. Sb. Math.
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S. A. Stepin, “On the resonance effect for singularly perturbed operator bundles”, Russian Math. Surveys, 51:5 (1996), 1003–1005
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