{"paper":{"title":"Generic continuous spectrum for multi-dimensional quasi periodic Schr\\\"odinger operators with rough potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Fan Yang, Rui Han","submitted_at":"2017-12-05T17:43:51Z","abstract_excerpt":"We study the multi-dimensional operator $(H_x u)_n=\\sum_{|m-n|=1}u_{m}+f(T^n(x))u_n$, where $T$ is the shift of the torus $\\T^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $\\alpha$ and generic continuous potentials. When $d\\geq 3$, the same result holds for frequencies under an explicit arithmetic criterion. We also show that general multi-dimensional operators with measurable potentials do not have eigenvalue for generic $\\alpha$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.01782","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"}