标题： 一维准周期连续薛定谔方程中的弹道输运 显示英文标题

标题： Ballistic Transport in One-Dimensional Quasi-Periodic Continuous Schrödinger Equation

摘要： 对于一维连续薛定谔方程 $${\rm i}\partial_t{q}(x,t)=-\partial_x^2 q(x,t) + V(\omega x) q(x,t), \quad x\in{\Bbb R},$$ 的解 $q(t)$，其中 $\omega\in{\Bbb R}^d$满足丢番图条件，且 $V$是 ${\Bbb T}^d$上的实解析函数，我们研究了扩散范数 $\|q(t)\|_{D}:=\left(\int_{\Bbb R}x^2|q(x,t)|^2dx\right)^{\frac12}$ 的增长速度，对于任意非零初始条件 $q(0)\in H^{1}({\Bbb R})$（具有 $\|q(0)\|_D<\infty$）。 我们证明了如果 $V$ 足够小，那么 $\|q(t)\|_{D}$ 随 $t$ 增长 {\it 线性地}。 显示英文摘要

摘要： For the solution $q(t)$ to the one-dimensional continuous Schr\"odinger equation $${\rm i}\partial_t{q}(x,t)=-\partial_x^2 q(x,t) + V(\omega x) q(x,t), \quad x\in{\Bbb R},$$ with $\omega\in{\Bbb R}^d$ satisfying a Diophantine condition, and $V$ a real-analytic function on ${\Bbb T}^d$, we consider the growth rate of the diffusion norm $\|q(t)\|_{D}:=\left(\int_{\Bbb R}x^2|q(x,t)|^2dx\right)^{\frac12}$ for any non-zero initial condition $q(0)\in H^{1}({\Bbb R})$ with $\|q(0)\|_D<\infty$. We prove that $\|q(t)\|_{D}$ grows {\it linearly} with $t$ if $V$ is sufficiently small.