Eigenvalue asymptotics of perturbed self-adjoint operators - Shkalikov, A. A. (Methods of Functional Analysis and Topology, 2012)
We study perturbations of a self-adjoint positive operator T, provided that a perturbation operator B satisfies the "local" subordinate condition \|Bφ_k\| bμ_k^β with some β0. Here {φ_k}_k=1^∞ is an orthonormal system of the eigenvectors of the operator T corresponding to the eigenvalues {μ_k}_k=1^∞. We introduce the concept of α-non-condensing sequence and prove the theorem on the comparison of the eigenvalue-counting functions of the operators T and T+B. Namely, it is shown that if {μ_k} is α-non-condensing then |n(r,T)- n(r, T+B)| C[ n(r+ar^γ, T) - n(r-ar^γ, T) ] +C_1 with some constants C, C_1, a and γ = (0, β, 2β +α -1)[0,1).