On some questions of spectral theory for differential equations whose coefficients have singular points (Q754056)

Using a variational method the author studies the properties of the eigenvalues and eigenfunctions of a second order differential operator whose coefficients tend to zero or to infinity in the boundary points. The case of so-called weak degeneration is considered. With the help of the theory of weighted functional spaces it is shown that in this case the basic results of the variational spectral theory of Hilbert-Sobolev remain true. The extremal properties of the eigenvalues and the completeness of the corresponding eigenfunctions are proved for problems with and without determined boundary conditions. The basic results of the paper had been published without proofs in the author's paper [Sov. Math., Dokl. 28, 31-36 (1983); translation from Dokl. Akad. Nauk SSSR 271, 37-42 (1983; Zbl 0556.34018)].