{"paper":{"title":"Control for Schr\\\"odinger equations on 2-tori: rough potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.AP","authors_text":"Jean Bourgain, Maciej Zworski, Nicolas Burq","submitted_at":"2013-01-07T17:43:25Z","abstract_excerpt":"For the Schr\\\"odinger equation, $ (i \\partial_t + \\Delta) u = 0 $ on a torus, an arbitrary non-empty open set $ \\Omega $ provides control and observability of the solution: $ \\| u |_{t = 0} \\|_{L^2 (\\T^2)} \\leq K_T \\| u \\|_{L^2 ([0,T] \\times \\Omega)} $. We show that the same result remains true for $ (i \\partial_t + \\Delta - V) u = 0 $ where $ V \\in L^2 (\\T^2) $, and $ \\T^2 $ is a (rational or irrational) torus. That extends the results of \\cite{AM}, and \\cite{BZ4} where the observability was proved for $ V \\in C (\\T^2) $ and conjectured for $ V \\in L^\\infty (\\T^2) $. The higher dimensional ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.1282","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"}