Bounds on the density of states of random Schrödinger operators

Abstract

Bounds are obtained on the unintegrated density of states ρ(E) of random Schrödinger operatorsH=−Δ + V acting onL ²(ℝ^d) orl ²(ℤ^d). In both cases the random potential is

$$V: = \sum\limits_{y \in \mathbb{Z}^d } {V_y \chi (\Lambda (y))}$$

in which the$\left\{ {V_y } \right\}_{y \in \mathbb{Z}^d }$ areIID random variables with densityf. The χ denotes indicator function, and in the continuum case the$\left\{ {\Lambda (y)} \right\}_{y \in \mathbb{Z}^d }$ are cells of unit dimensions centered ony∈ℤ^d. In the finite-difference case Λ(y) denotes the sitey∈ℤ^d itself. Under the assumptionf ∈ L ^1+ɛ₀ (ℝ) it is proven that in the finitedifference casep ∈ L ^∞(ℝ), and that in thed= 1 continuum casep ∈ L ^∞_loc (ℝ).