zbMATH — the first resource for mathematics

Dilatation and the asymptotics of the eigenvalues of spectral problems with singularities. (English. Russian original) Zbl 0628.35077
Funct. Anal. Appl. 20, 277-281 (1986); translation from Funkts. Anal. Prilozh. 20, No. 4, 29-34 (1986).
The eigenvalue problem \(Au=\lambda^ 2\mu u\) is investigated providing that A is a uniformly elliptic selfadjoint operator of the 2nd order and \(\mu\) is a nonnegative weight function. The term singularity means either the nonregularity of \(\mu\) or of the coefficients of A, degeneration of \(\mu\), unboundedness of the domain or nonregularity of the boundary. It is shown how the method of dilatations facilitates to derive very precise asymptotics of eigenvalues of the spectral problem with sufficiently weak singularities.
Reviewer: P.Burda

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
Full Text: DOI
[1] V. Ya. Ivrii, ”Weyl’s asymptotic formula for the Laplace?Beltrami operator in Riemann polyhedra and in domains with conical singularities of the boundary,” Dokl. Akad. Nauk SSSR,38, No. 1, 35-38 (1986).
[2] G. V. Rozenblyum, ”Distribution of the discrete spectrum of singular differential operators,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 1, 75-86 (1976).
[3] I. P. Kornfel’d, Ya. G. Sinai and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).
[4] L. Hörmander, ”The spectral function of an elliptic operator,” Acta Math.,121, No. 1-2, 193-218 (1968). · Zbl 0164.13201 · doi:10.1007/BF02391913
[5] V. Ivrii, ”Precise Spectral Asymptotics for Elliptic Operators,” Lecture Notes in Math., No. 1100, Springer, Berlin (1984).
[6] V. Ya. Ivrii, ”On the exact spectral asymptotics for the Laplace?Beltrami operator under general elliptic boundary conditions,” Funkts. Anal. Prilozhen.,15, No. 1, 74-75 (1981). · Zbl 0502.35068 · doi:10.1007/BF01082390
[7] M. A. Shubin, Pseudodifferential Operators and Spectral Theory [in Russian], Nauka, Moscow (1978). · Zbl 0451.47064
[8] V. Ya. Ivrii, ”On the second term of the spectral asymptotics for the Laplace?Beltrami operator on manifolds with boundary,” Funkts. Anal. Prilozhen.,14, No. 2, 25-34 (1980).
[9] D. G. Vasil’ev, Two-term asymptotics of the spectrum of a boundary-value problem with a piecewise-smooth boundary,” Usp. Mat. Nauk,40, No. 5, 233 (1985).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.