REGULARITY OF THE INVARIANT MEASURE AND OF THE DENSITY OF STATES IN THE ONE-DIMENSIONAL ANDERSON MODEL

Let H = - 1 2 Δ + V on l2(Z), where V(x), xε{lunate}Z are i.i.d.r.v.'s with common probability distribution μ. Let h(t) = ∝ e-itvdμ(v), let N(E) be the integrated density of states, and let νE be the invariant measure at energy E. It is proven: 1. 1. Let h be d-times differentiable on (0, + ∞) with bounded derivatives such that h(j)(t) → 0 as t → +∞ for j = 0, 1, ..., d. Then: (i) N(E) is a Cd function. (ii) If d ≥ 3, νE is absolutely continuous for all Eε{lunate}R and dνE dx is of class C[ (d - 1) 2] - 1jointly in E and x. 2. 2. Let h be differentiable on (0, +∞) with h(j)(t) = O(e-b|t|) for j = 0, 1 and some b > 0. Then dνE dx is jointly analytic in E and x. © 1990.