Spectral type of a class of random Jacobi operators

Abstract

In this paper, we use the generalized Prüfer variables to study the spectral type of a class of random Jacobi operators (Hτ,ωλu)(n)=τnu(n+1)+τn−1u(n−1)+λanωnu(n), in which the decay speed of the parameters an is n−α for some α > 0. We will show that the operator has an absolutely continuous spectrum for α>12, a pure point spectrum for 0<α<12, and a transition from a singular continuous spectrum to a pure point spectrum in α=12.