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arXiv · 1507.08909

Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation

Zhiyan Zhao
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Abstract

For the solution $q(t)=(q_n(t))_{n\in\mathbb Z}$ to one-dimensional discrete Schrödinger equation $${\rm i}\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(θ+nω) q_n, \quad n\in\mathbb Z,$$ with $ω\in\mathbb R^d$ 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 $θ\in\mathbb T^d$ if $V$ is sufficiently small.

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Zhiyan Zhao. 2016-02-10. Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation. https://doi.org/10.1007/s00220-016-2605-z

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