{"paper":{"title":"Density of the spectrum of Jacobi matrices with power asymptotics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Raphael Pruckner","submitted_at":"2017-04-22T12:20:31Z","abstract_excerpt":"We consider Jacobi matrices $J$ whose parameters have the power asymptotics $\\rho_n=n^{\\beta_1} \\left( x_0 + \\frac{x_1}{n} + {\\rm O}(n^{-1-\\epsilon})\\right)$ and $q_n=n^{\\beta_2} \\left( y_0 + \\frac{y_1}{n} + {\\rm O}(n^{-1-\\epsilon})\\right)$ for the off-diagonal and diagonal, respectively. We show that for $\\beta_1 > \\beta_2$, or $\\beta_1=\\beta_2$ and $2x_0 > |y_0|$, the matrix $J$ is in the limit circle case and the convergence exponent of its spectrum is $1/\\beta_1$. Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix $J$ have a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06789","kind":"arxiv","version":2},"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"}