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We examine the selfadjoint operator in . We assume that the potential tends to as . Under these conditions the spectrum of is discrete. In the paper the well-known asymptotic formula (*)for the distribution function of the eigenvalues is justified under very weak assumptions on , namely the following conditions: 1) , where . 2) almost everywhere when . 3) There exist a continuous function , , , and an index such that for any , , .Bibliography: 12 items.

This paper is devoted to the investigation of the spectrum of a polyharmonic operator in unbounded domains. The class of domains for which the spectrum of the corresponding first boundary value problem is discrete is examined. The classical asymptotic formula for eigenvalues is extended to the case of domains of finite volume. A two-sided bound for the distribution function of the eigenvalues is obtained in the general case. If the domain behaves sufficiently regularly at infinity, then the upper and lower bounds coincide in order. The results are new also for the Laplace operator. Bibliography: 13 items.