Abstract
The Bethe strip of width m is the cartesian product \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {B}\times \lbrace 1,\ldots ,m\rbrace$\end{document}, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {B}$\end{document} 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_\lambda =\frac{1}{2} \Delta \otimes 1 + 1 \otimes A,+,\lambda \mathcal {V}$\end{document} on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {V}$\end{document} is a random matrix potential, and λ is the disorder parameter. Given any closed interval \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$I\subset \big (!-!\sqrt{K}+a_{{\rm max}},\sqrt{K}+a_{\rm {min}}\big )$\end{document}, where a~min~ and a~max~ are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H~λ~ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.