Ballistic behavior for random Schrödinger operators on the Bethe strip

Abstract

The Bethe strip of width $m$ is the Cartesian product $\mathbb{B}\times \{1,\dots ,m\}$, where $\mathbb{B}$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $H_{\lambda }=\frac{1}{2}\Delta \otimes 1+1\otimes A+\lambda \mathcal{V}$ on a Bethe strip with connectivity $K\ge 2$, where $A$ is an $m\times m$ symmetric matrix, $\mathcal{V}$ is a random matrix potential, and $\lambda$ is the disorder parameter. Under certain conditions on $A$ and $K$, for which we previously proved the existence of absolutely continuous spectrum for small $\lambda$, we now obtain ballistic behavior for the spreading of wave packets evolving under $H_{\lambda }$ for small $\lambda$.