Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs

The subject of this work is random Schrödinger operators on regular rooted tree graphs [script T sign] with stochastically homogeneous disorder. The operators are of the form H _λ (ω)=T+U+λ V(ω) acting in l²(script T sign), with T the adjacency matrix, U a radially periodic potential, and V(ω) a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of H _λ (ω) exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder in case U=0 and V(ω) given by iid random variables on script T sign. Our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder. The method yields the continuity of the ac spectral density of H _λ (ω) at λ=0. The latter is shown to converge in the L ¹-sense over closed Borel sets in which H ₀ has no singular spectrum. The analysis extends to random potentials whose values at different sites need not be independent, assuming only that their joint distribution is weakly correlated across different tree branches.