{"paper":{"title":"A family of anisotropic integral operators and behaviour of its maximal eigenvalue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"A.V. Sobolev, B.S Mityagin","submitted_at":"2011-06-01T09:38:26Z","abstract_excerpt":"We study the family of compact integral operators $\\mathbf K_\\beta$ in $L^2(\\mathbb R)$ with the kernel K_\\beta(x, y) = \\frac{1}{\\pi}\\frac{1}{1 + (x-y)^2 + \\beta^2\\Theta(x, y)}, depending on the parameter $\\beta >0$, where $\\Theta(x, y)$ is a symmetric non-negative homogeneous function of degree $\\gamma\\ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_\\beta$ of $\\mathbf K_\\beta$: M_\\beta = 1 - \\lambda_1 \\beta^{\\frac{2}{\\gamma+1}} + o(\\beta^{\\frac{2}{\\gamma+1}}), \\beta\\to 0, where $\\lambda_1$ is the lowest eigenvalue of the operator $\\mathbf A = |d/dx| + "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0127","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"}