Eigenvalue Asymptotics of Perturbed Self-adjoint Operators

Details Eigenvalue Asymptotics of Perturbed Self-adjoint Operators Eigenvalue Asymptotics of Perturbed Self-adjoint Operators

2012-02-23

arxiv.org/abs/1202.5204v1

Mathematics

Spectral Theory

12 pages

Scientific paper

We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies "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 the difference of the eigenvalue-counting functions is subject to relation $$|n(r,\, T)- n(r,\, T+B)| \leqslant C[n(r+ar^\gamma,\, T) - n(r-ar^\gamma,\, T)] +C_1 $$ with some constants $C, C_1, a$ and $\gamma = \max(0, \beta, 2\beta+\alpha-1)\in [0,1)$.

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