Absolutely Continuous Spectrum of One Random Elliptic Operator

In dimension d ≥ 5, we consider the differential operator (1.1) H0 = −∆+ τζ(x)|x|(−∆θ), ε > 0, τ > 0, where ∆θ is the Laplace-Beltrami operator on the unit sphere S = {x ∈ R : |x| = 1} and ζ is the characteristic function of the complement to the unit ball {x ∈ R : |x| ≤ 1}. The standard argument with separation of variables allows one to define this operator as the orthogonal sum of one-dimensional Schrödinger operators, which implies that H0 is essentially self-adjoint on C∞ 0 (R). The spectrum of this operator has an absolutely continuous component, which coincides with the positive half-line [0,∞) as a set. We perturb now the operator H0 by a real valued potential V = Vω that depends on the random parameter ω = {ωn}n∈Zd :