{"paper":{"title":"Long time growth of Sobolev norms in time dependent semiclassical anharmonic oscillators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Alberto Maspero, Emanuele Haus","submitted_at":"2019-04-07T18:19:26Z","abstract_excerpt":"We consider the semiclassical Schr\\\"odinger equation on $\\mathbb R^d$ given by $$\\mathrm{i} \\hbar \\partial_t \\psi = \\left(-\\frac{\\hbar^2}{2} \\Delta + W_l(x) \\right)\\psi + V(t,x)\\psi ,$$ where $W_l$ is an anharmonic trapping of the form $W_l(x)= \\frac{1}{2l}\\sum_{j=1}^d x_j^{2l}$, $l\\geq 2$ is an integer and $\\hbar$ is a semiclassical small parameter. We construct a smooth potential $V(t,x)$, bounded in time with its derivatives, and an initial datum such that the Sobolev norms of the solution grow at a logarithmic speed for all times of order $\\log^{\\frac12}(\\hbar^{-1})$. The proof relies on t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03703","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"}