The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

Given two self-adjoint operators A and V = V + − V − , we study the motion of the eigenvalues of the operator A t = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N + V λ α , N − V λ α , and N0 V λ α defined as the number of eigenvalues of the operator A t that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α→∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys. 193 (1998), 233–243] are extended and given new applications to differential operators.