{"paper":{"title":"Eigenvalue Asymptotics of Perturbed Self-adjoint Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"A. A. Shkalikov","submitted_at":"2012-02-23T15:11:01Z","abstract_excerpt":"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 $\\alpha-$n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5204","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}