{"paper":{"title":"Long time dynamics of Schr\\\"odinger and wave equations on flat tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Maspero, Massimiliano Berti","submitted_at":"2018-11-16T09:13:43Z","abstract_excerpt":"We consider a class of linear time dependent Schr\\\"odinger equations and quasi-periodically forced nonlinear Hamiltonian wave/Klein Gordon and Schr\\\"odinger equations on arbitrary flat tori. For the linear Schr\\\"odinger equation, we prove a $t^\\epsilon$ $(\\forall \\epsilon >0)$ upper bound for the growth of the Sobolev norms as the time goes to infinity. For the nonlinear Hamiltonian PDEs we construct families of time quasi-periodic solutions. Both results are based on \"clusterization properties\" of the eigenvalues of the Laplacian on a flat torus and on suitable \"separation properties\" of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06714","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"}