Ballistic Behavior for Random Schr\"odinger Operators on the Bethe Strip

TL;DR

This paper proves ballistic wave packet spreading for certain random Schrödinger operators on the Bethe strip, extending previous spectral results to dynamical behavior under small disorder.

Contribution

It establishes ballistic transport for Anderson-like Hamiltonians on the Bethe strip, building on prior spectral analysis to demonstrate wave packet spreading.

Findings

Ballistic behavior occurs for small disorder parameter 

Absolutely continuous spectrum is linked to wave packet spreading

Results apply to Hamiltonians with specific matrix conditions

Abstract

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $H_\lambda=\frac12 \Delta \otimes 1 + 1 \otimes A+\lambda \Vv$ on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, $\Vv$ 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 $\lambda$.