Extended States in the Anderson Model on the Bethe Lattice

Abstract

We prove that the Anderson HamiltonianHλ=-Δ+λVon the Bethe lattice has "extended states" for small disorder. More precisely, given any closed intervalIcontained in the interior of the spectrum of the Laplacian on the Bethe lattice, we prove that for small disorderHλhas purely absolutely continuous spectrum inIwith probability one (i.e.,σac(Hλ)∩I=Iandσ pp(Hλ)∩I=σsc(H λ)∩I= with probability one), and its integrated density of states is continuously differentiable on the intervalI. © 1998 Academic Press.