Finite-dimensional perturbations of self-adjoint operators

Abstract

We study finite-dimensional perturbationsA+γB of a self-adjoint operatorA acting in a Hilbert space\(\mathfrak{H}\). We obtain asymptotic estimates of eigenvalues of the operatorA+γB in a gap of the spectrum of the operatorA as γ → 0, and asymptotic estimates of their number in that gap. The results are formulated in terms of new notions of characteristic branches ofA with respect to a finite-dimensional subspace of\(\mathfrak{H}\) on a gap of the spectrum σ(A) and asymptotic multiplicities of endpoints of that gap with respect to this subspace. It turns out that ifA has simple spectrum then under some mild conditions these asymptotic multiplicities are not bigger than one. We apply our results to the operator(Af)(t)=tf(t) onL2([0, 1],ρc), whereρc is the Cantor measure, and obtain the precise description of the asymptotic behavior of the eigenvalues ofA+γB in the gaps of\(\sigma (A) = \mathfrak{C}\)(= the Cantor set).