{"paper":{"title":"Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Zhiyan Zhao","submitted_at":"2015-07-31T15:09:38Z","abstract_excerpt":"For the solution $q(t)=(q_n(t))_{n\\in\\mathbb Z}$ to one-dimensional discrete Schr\\\"odinger equation $${\\rm i}\\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(\\theta+n\\omega) q_n, \\quad n\\in\\mathbb Z,$$ with $\\omega\\in\\mathbb R^d$ Diophantine, and $V$ a small real-analytic function on $\\mathbb T^d$, we consider the growth rate of the diffusion norm $\\|q(t)\\|_{D}:=\\left(\\sum_{n}n^2|q_n(t)|^2\\right)^{\\frac12}$ for any non-zero $q(0)$ with $\\|q(0)\\|_{D}<\\infty$. We prove that $\\|q(t)\\|_{D}$ grows {\\it linearly} with the time $t$ for any $\\theta\\in\\mathbb T^d$ if $V$ is sufficiently small."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08909","kind":"arxiv","version":4},"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"}