Title: Ballistic Transport for Discrete Multi-Dimensional Schrödinger Operators With Decaying Potential Show Chinese title

Title: 离散多维薛定谔算子带衰减势的弹道输运

Abstract: We consider the discrete Schr\"odinger operator $H = -\Delta + V$ on $\ell^2(\mathbb{Z}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in\mathbb{N}^*$, where $\Delta$ is the standard discrete Laplacian and $V_n = o(|n|^{-1})$ as $|n| \to \infty$. We prove that the unitary evolution $e^{-i tH}$ exhibits ballistic transport in the sense that, for any $r > 0$, the weighted $\ell^2-$norm $$\|e^{-i tH}u\|_r:=\left(\sum_{n\in\mathbb{Z}^d} (1+|n|^2)^{r} |(e^{-i tH}u)_n|^2\right)^\frac12 $$ grows at rate $\simeq t^r$ as $t\to \infty$, provided that the initial state $u$ is in the absolutely continuous subspace and satisfies $\|u\|_r<\infty$. The proof relies on commutator methods and a refined Mourre estimate, which yields quantitative lower bounds on transport for operators with purely absolutely continuous spectrum over appropriate spectral intervals. Compactness arguments and localized spectral projections are used to extend the result to perturbed operators, extending the classical result for the free Laplacian to a broader class of decaying potentials. Show Chinese abstract

Abstract: 我们考虑在任意格点维数$d\in\mathbb{N}^*$下的离散薛定谔算子$H = -\Delta + V$在$\ell^2(\mathbb{Z}^d)$上带有衰减势，其中$\Delta$是标准离散拉普拉斯算子，$V_n = o(|n|^{-1})$作为$|n| \to \infty$。 我们证明了幺正演化 $e^{-i tH}$在某种意义上表现出弹道输运，即对于任何 $r > 0$，加权 $\ell^2-$范数 $$\|e^{-i tH}u\|_r:=\left(\sum_{n\in\mathbb{Z}^d} (1+|n|^2)^{r} |(e^{-i tH}u)_n|^2\right)^\frac12 $$随 $t\to \infty$以速率 $\simeq t^r$增长，前提是初始状态 $u$位于绝对连续子空间中并满足 $\|u\|_r<\infty$。 证明依赖于换位子方法和改进的Mourre估计，这为在适当谱区间上具有纯绝对连续谱的算子提供了传输的定量下界。 紧致性论证和局部谱投影被用来将结果扩展到受扰算子，将自由拉普拉斯算子的经典结果扩展到更广泛的衰减势类。