Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

Abstract

The Bethe Strip of width m is the cartesian product B × {1, . . . , m}, where 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 Hλ = 1/2∆⊗1+ 1⊗A + λV on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, V is a random matrix potential, and λ is the disorder parameter. Given any closed interval I ⊂ (−√K + amax,√K + amin), where amin and amax are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator Hλ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.