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

We consider the discrete Schrödinger operator $H=-\mathrm{\Delta }+V$ on ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$ with a decaying potential, in arbitrary lattice dimension $d\in {\mathbb{N}}^{\ast }$, where $\mathrm{\Delta }$ is the standard discrete Laplacian and $V_n=o(\mathrm{\mid }n{\mathrm{\mid }}^{-1})$ as $\mathrm{\mid }n\mathrm{\mid }\to \mathrm{\infty }$. %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 ${\mathrm{\ell }}^2-$norm $$\mathrm{\parallel }{e}^{-itH}u{\mathrm{\parallel }}_r:={(\sum _{n\in {\mathbb{Z}}^d}(1+\mathrm{\mid }n{\mathrm{\mid }}^2{)}^r\mathrm{\mid }(e^{-itH}u{)}_n{\mathrm{\mid }}^2)}^{\frac{1}{2}}$$ grows at rate $\simeq t^r$ as $t\to \mathrm{\infty }$, provided that the initial state $u$ is in the absolutely continuous subspace and satisfies $\mathrm{\parallel }u{\mathrm{\parallel }}_r<\mathrm{\infty }$. 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.