Ballistic Transport for Discrete Multi-Dimensional Schrödinger Operators With Decaying Potential
David Damanik, Zhiyan Zhao
TL;DR
This work proves ballistic transport for discrete multidimensional Schrödinger operators with decaying potentials, showing that the unitary evolution e^{-itH} drives spreading in weighted ℓ^2-norms at rate t^r for any r>0, provided the initial state lies in the absolutely continuous subspace. The authors combine commutator techniques with a refined Mourre estimate to obtain a robust order-1 ballistic lower bound and prove an optimal upper bound, handling perturbations by compact operators and extending the free Laplacian result to potentials with decay V_n = o(|n|^{-1}). They also establish absence of singular continuous spectrum and identify the ac spectrum as [-2d,2d] in the decaying-potential setting. The analysis advances understanding of transport in discrete quantum systems and delineates precise decay thresholds that guarantee ballistic dynamics in higher dimensions.
Abstract
We consider the discrete Schrödinger operator $H = -Δ+ V$ on $\ell^2(\mathbb{Z}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in\mathbb{N}^*$, where $Δ$ 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.