Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

The Bethe Strip of width $m$ is the cartesian product $\text{B}\times \{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$, where $\text{B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson-like Hamiltonians $;H_\lambda=\frac12 \Delta \otimes 1 + 1 \otimes A + \lambda \Vv$onaBethestripwithconnectivity$K \geq 2$,where$A$isan$m\times m$symmetricmatrix,$\Vv$isarandommatrixpotential,and$\lambda$isthedisorderparameter\mathrm{.}Givenanyclosedinterval$I\subset (-\sqrt{K}+a_{$\mathrm{m}\mathrm{a}\mathrm{x}$},\sqrt{K}+a_{$\mathrm{m}\mathrm{i}\mathrm{n}$})$,where$a_{$\mathrm{m}\mathrm{i}\mathrm{n}$}$and$a_{$\mathrm{m}\mathrm{a}\mathrm{x}$}$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $\lambda$ small the random Schr"odinger operator $H_{\lambda }$ has purely absolutely continuous spectrum in $I$ with probability one and its integrated density of states is continuously differentiable on the interval $I$.