{"paper":{"title":"Resonances for open quantum maps and a fractal uncertainty principle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP","nlin.CD"],"primary_cat":"math.SP","authors_text":"Long Jin, Semyon Dyatlov","submitted_at":"2016-08-07T16:22:37Z","abstract_excerpt":"We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions $\\delta\\in (0,1)$. We show that the size of the spectral gap is strictly greater than the standard bound $\\max(0,{1\\over 2}-\\delta)$ for all values of $\\delta$, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02238","kind":"arxiv","version":3},"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"}