Finite-dimensional perturbations of self-adjoint operators

We study finite-dimensional perturbations A + gamma B of a self adjoint ope rator A acting in a Hilbert space h. We obtain asymptotic estimates of eige nvalues of the operator A + gamma B in a gap of the spectrum of the operato r A as gamma --> 0, and asymptotic estimates of their number in that gap. T he results are formulated in terms of new notions of characteristic branche s of A with respect to a finite-dimensional subspace of h on a gap of the s pectrum sigma(A) and asymptotic multiplicities of endpoints of that gap wit h 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 L-2([0, 1 ], rho(c)), where rho(c) is the Canter measure, and obtain the precise desc ription of the asymptotic behavior of the eigenvalues of A + gamma B in the gaps of sigma(A) = C(= the Canter set).