Finite-dimensional perturbations of self-adjoint operators

We study finite-dimensional perturbations A + γB of a self-adjoint operator A acting in a Hilbert space Heng hooktop sign. We obtain asymptotic estimates of eigenvalues of the operator A+γB in a gap of the spectrum of the operator A as γ → 0, and asymptotic estimates of their number in that gap. The results are formulated in terms of new notions of characteristic branches of A with respect to a finite-dimensional subspace of Heng hooktop sign 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 if A 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) on L²([0, 1],p_c), where p_c is.the Cantor measure, and obtain the precise description of the asymptotic behavior of the eigenvalues of A + γB in the gaps of σ(A) = script C sign(= the Cantor set).