Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Abstract

Let A be a self-adjoint elliptic second-order differential operator, let (α, β) be an inner gap in the spectrum of A, and let B(t) = A + tW * W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (α, β) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(λ, A, W, τ), λ ∈ (α, β), τ > 0 (the number of eigenvalues of the operator B(t) passing the point λ ∈ (α, β) as t increases from 0 to τ) is considered. Estimates of N(λ, A, W, τ) are obtained. For the perturbation W * W of a special form, the asymptotics of N(λ, A, W, τ) as τ → +∞ is given. Bibliography: 5 titles.