Ballistic behavior for random Schrödinger operators on the Bethe strip

Abstract

The Bethe strip of width $m$ is the Cartesian product $\mathbb B\times\{1,\ldots,m\}$, where $\mathbb B$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $\;H_\lambda=\frac12 \Delta \otimes 1 + {1 \otimes A}+\lambda \mathcal{V}$ 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 $\lambda$.