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 $(H_{\tau ,\omega }^{\lambda }u)(n)={\tau }_{n}u(n+1)+{\tau }_{n-1}u(n-1)+\lambda a_n{\omega }_{n}u(n),$ in which the decay speed of the parameters a_n is n^−α for some α > 0. We will show that the operator has an absolutely continuous spectrum for $\alpha >\frac{1}{2}$ , a pure point spectrum for $0<\alpha <\frac{1}{2}$ , and a transition from a singular continuous spectrum to a pure point spectrum in $\alpha =\frac{1}{2}$ .