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

Abstract

We consider the discrete Schrödinger operator $H=-\hat{I}”+V$ on ${\ell }^2({\mathbb{Z}}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in {\mathbb{N}}^{\ast }$, where $\hat{I}”$ is the standard discrete Laplacian and $V_n=o(|n{|}^{-1})$ as $|n|\to \infty$. %We prove the absence of singular continuous spectrum for $H$. For the unitary evolution $e^{-itH}$, we prove that it exhibits ballistic transport in the sense that, for any $r>0$, the weighted ${\ell }^2-$norm $$\Vert e^{-itH}u{\Vert }_r:={\left(\sum _{n\in {\mathbb{Z}}^d}(1+|n{|}^2{)}^r|(e^{-itH}u{)}_n{|}^2\right)}^{\frac{1}{2}}$$ grows at rate $\simeq t^r$ as $t\to \infty$, provided that the initial state $u$ is in the absolutely continuous subspace and satisfies $\Vert u{\Vert }_r<\infty$. The proof relies on commutator methods and Mourre estimate, which yields quantitative lower bounds on transport for operators with purely absolutely continuous spectrum over appropriate spectral intervals.

19 pages