Eigenvalue asymptotics of perturbed self-adjoint operators
Abstract
We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies the "local" subordinate condition $\|B\varphi_k\| \leqslant b\mu_k^{\beta}$ with some $\beta<1$ and $b>0$. Here $\{\varphi_k\}_{k=1}^\infty$ is an orthonormal system of the eigenvectors of the operator $T$ corresponding to the eigenvalues $\{\mu_k\}_{k=1}^\infty$. We introduce the concept of $\alpha$-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 $\{\mu_k\}$ is $\alpha-$non-condensing then $$ |n(r,T)- n(r, T+B)| \leqslant C\left[ n(r+ar^\gamma,\, T) - n(r-ar^\gamma,\, T) \right] +C_1 $$ with some constants $C, C_1, a$ and $\gamma = \max(0,\, \beta,\, 2\beta +\alpha -1)\in [0,1)$.
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