Paper detail

Eigenvalue Asymptotics of Perturbed Self-adjoint Operators

We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies "local" subordinate condition $\|Bφ_k\|\leqslant bμ_k^β$ with some $β<1$ and $b>0$. Here $\{φ_k\}_{k=1}^\infty$ is an orthonormal system of the eigenvectors of the operator $T$ corresponding to the eigenvalues $\{μ_k\}_{k=1}^\infty$. 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 the difference of the eigenvalue-counting functions is subject to relation $$|n(r,\, T)- n(r,\, T+B)| \leqslant C[n(r+ar^γ,\, T) - n(r-ar^γ,\, T)] +C_1 $$ with some constants $C, C_1, a$ and $γ= \max(0, β, 2β+α-1)\in [0,1)$.