Ballistic Transport for Discrete Multi-Dimensional Schrödinger Operators With Decaying Potential

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.