Stepin, S. A. Spectrum and completeness of natural oscillations of the atmosphere with temperature stratification. (English. Russian original) Zbl 0816.76023 Russ. Acad. Sci., Sb., Math. 79, No. 1, 179-190 (1994); translation from Mat. Sb. 184, No. 6, 83-98 (1993). Summary: 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 in the boundary condition. In this article this problem in 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. MSC: 76B60 Atmospheric waves (MSC2010) 86A10 Meteorology and atmospheric physics 34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators 47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) Keywords:nonselfadjoint singular boundary eigenvalue problem; spectral parameter; operator pencil; positive compact operators; energy quadratic form; multiplicity; Riesz bases PDFBibTeX XMLCite \textit{S. A. Stepin}, Russ. Acad. Sci., Sb., Math. 79, No. 1, 83--98 (1993; Zbl 0816.76023); translation from Mat. Sb. 184, No. 6, 83--98 (1993) Full Text: DOI