{"paper":{"title":"Homogeneity of the spectrum for quasi-periodic Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"David Damanik, Michael Goldstein, Mircea Voda, Wilhelm Schlag","submitted_at":"2015-05-19T08:25:10Z","abstract_excerpt":"We consider the one-dimensional discrete Schr\\\"odinger operator $$ \\bigl[H(x,\\omega)\\varphi\\bigr](n)\\equiv -\\varphi(n-1)-\\varphi(n+1) + V(x + n\\omega)\\varphi(n)\\ , $$ $n \\in \\mathbb{Z}$, $x,\\omega \\in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,\\omega)>0$ for all $E$. Let $\\mathcal{S}_\\omega$ be the spectrum of $H(x,\\omega)$. For all $\\omega$ obeying the Diophantine condition $\\omega \\in \\mathbb{T}_{c,a}$, we show the following: if $\\mathcal{S}_\\omega \\cap (E',E\")\\neq \\emptyset$, then $\\mathcal{S}_\\omega \\cap (E',E\")$ is homogeneous in the sense of Carleson (see [Car83]). Furtherm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04904","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"}