Spectrum and completeness of natural oscillations of the atmosphere with temperature stratification

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.