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 "local" subordinate condition $\Vert B{\phi }_k\Vert ⩽b{\mu }_k^{\beta }$ with some $\beta <1$ and $b>0$. Here $\{{\phi }_k{\}}_{k=1}^{\infty }$ 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 the difference of the eigenvalue-counting functions is subject to relation $$|n(r,T)-n(r,T+B)|⩽C[n(r+a{r}^{\gamma },T)-n(r-a{r}^{\gamma },T)]+C_1$$ with some constants $C,C_1,a$ and $\gamma =\max (0,\beta ,2\beta +\alpha -1)\in [0,1)$.

12 pages

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