A Note on the Analyticity of Density of States

Abstract

We consider the d-dimensional Anderson model, and we prove the density of states is locally analytic if the single site potential distribution is locally analytic and the disorder is large. We employ the random walk expansion of resolvents and a simple complex function theory trick. In particular, we discuss the uniform distribution case, and we obtain a sharper result using more precise computations. The method can be also applied to prove the analyticity of the correlation functions.

Appendix

We define the correlation function K(e ₁,e ₂) here. Given a pair of vectors f,g∈ℓ ²(ℤ^d) and any self adjoint operator A, we see that the limits

exist for almost every λ with respect to the Lebesgue measure, by using Theorem 1.4.16 [8] and polarization identity to write the finite complex measure 〈f,E _A (⋅)g〉 as a linear combination of the positive finite measures

Thus the function r(E)=〈f,δ(H−E)g〉 gives the density of the measure 〈f,E _A (⋅)g〉 almost every E. Similarly computing (and justifying the exchange of ${\mathbb{E}}$ and the integrals with respect to λ ₁,λ ₂, by Fubini),

(5)

where μ is the finite complex correlation measure

As in the proof of Theorem 1.4.16 [8], we can show that the limits

exist for almost every (e ₁,e ₂)∈ℝ². Now note that the complex valued function

of two complex variables is analytic in a neighborhood of (e ₁,e ₂) if and only if it is analytic as a function of the real variables (e ₁,e ₂). Therefore in the proof of Theorem 4 we consider the function

and show it is analytic in the region stated in the theorem.