{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:AILUMKJ2GJ6PPXKDRC73A2ITCM","short_pith_number":"pith:AILUMKJ2","schema_version":"1.0","canonical_sha256":"021746293a327cf7dd4388bfb0691313393c50cb2ecc04df0041edc0af598c23","source":{"kind":"arxiv","id":"1512.02195","version":2},"attestation_state":"computed","paper":{"title":"Ballistic Transport and Absolute Continuity of One-Frequency Schr\\\"{o}dinger Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DS","authors_text":"Zhiyan Zhao, Zhiyuan Zhang","submitted_at":"2015-12-07T20:32:13Z","abstract_excerpt":"For the solution $u(t)$ to the discrete Schr\\\"odinger equation $${\\rm i}\\frac{d}{dt}u_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(\\theta + n\\alpha)u_n(t), \\quad n\\in\\Z,$$ with $\\alpha\\in\\R\\setminus\\Q$ and $V\\in C^\\omega(\\T,\\R)$, we consider the growth rate with $t$ of its diffusion norm $\\langle u(t)\\rangle_{p}:=\\left(\\sum_{n\\in\\Z}(n^{p}+1) |u_n(t)|^2\\right)^\\frac12$, and the (non-averaged) transport exponents $$\\beta_u^{+}(p) := \\limsup_{t \\to \\infty} \\frac{2\\log \\langle u(t)\\rangle_{p}}{p\\log t}, \\quad \\beta_u^{-}(p):= \\liminf_{t \\to \\infty} \\frac{2\\log \\langle u(t)\\rangle_{p}}{p\\log t}.$$ We will show 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Transport and Absolute Continuity of One-Frequency Schr\\\"{o}dinger Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DS","authors_text":"Zhiyan Zhao, Zhiyuan Zhang","submitted_at":"2015-12-07T20:32:13Z","abstract_excerpt":"For the solution $u(t)$ to the discrete Schr\\\"odinger equation $${\\rm i}\\frac{d}{dt}u_n(t)=-(u_{n+1}(t)+u_{n-1}(t))+V(\\theta + n\\alpha)u_n(t), \\quad n\\in\\Z,$$ with $\\alpha\\in\\R\\setminus\\Q$ and $V\\in C^\\omega(\\T,\\R)$, we consider the growth rate with $t$ of its diffusion norm $\\langle u(t)\\rangle_{p}:=\\left(\\sum_{n\\in\\Z}(n^{p}+1) |u_n(t)|^2\\right)^\\frac12$, and the (non-averaged) transport exponents $$\\beta_u^{+}(p) := \\limsup_{t \\to \\infty} \\frac{2\\log \\langle u(t)\\rangle_{p}}{p\\log t}, \\quad \\beta_u^{-}(p):= \\liminf_{t \\to \\infty} \\frac{2\\log \\langle u(t)\\rangle_{p}}{p\\log t}.$$ We will show 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