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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 $\gamma = \max(0,\, \beta,\, 2\beta +\alpha -1)\in [0,1)$.

Article Information

 Title Eigenvalue asymptotics of perturbed self-adjoint operators Source Methods Funct. Anal. Topology, Vol. 18 (2012), no. 1, 79-89 MathSciNet MR2953332 zbMATH 1243.47032 Copyright The Author(s) 2012 (CC BY-SA)

Authors Information

A. A. Shkalikov
Department of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow, Russia

Citation Example

A. A. Shkalikov, Eigenvalue asymptotics of perturbed self-adjoint operators, Methods Funct. Anal. Topology 18 (2012), no. 1, 79-89.

BibTex

@article {MFAT634,
AUTHOR = {Shkalikov, A. A.},
TITLE = {Eigenvalue asymptotics of perturbed self-adjoint operators},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {18},
YEAR = {2012},
NUMBER = {1},
PAGES = {79-89},
ISSN = {1029-3531},
MRNUMBER = {MR2953332},
ZBLNUMBER = {1243.47032},
URL = {http://mfat.imath.kiev.ua/article/?id=634},
}