An Invitation to Random Schr¨odinger operators - FernUniversität in ...

43
∣ ∑
(H ΛL − z) −1 (n, n) − (H − z) −1 (n, n) ∣ n∈Λ L
= ∣ ∑ ∑
(H ΛL − z) −1 (n, k) (H − z) −1 (k ′ , n) ∣ n∈Λ L (k,k ′ )∈∂Λ L
k∈Λ L , k ′ ∈∁Λ L
∑
≤
|(H ΛL − z) −1 (n, k)| 2) 12 · ( ∑
|(H − z) −1 (k ′ , n)| 2) 1 2
=
( ∑
(k,k ′ )∈∂Λ L n
k∈Λ L , k ′ ∈∁Λ L
∑
||(H ΛL − z) −1 δ k || · ||(H − z) −1 δ k ′||
(k,k ′ )∈∂Λ L
k∈Λ L , k ′ ∈∁Λ L
≤ c L d−1 ||(H ΛL − z) −1 || · ||(H − z) −1 ||
≤
c
(Im z) 2 Ld−1 . (5.59)
Hence
n
∫
|
∫
r z (λ) d˜ν L (λ) −
r z (λ) dν L (λ)| ≤
c ′
(Im z) 2 · 1
L → 0 as L → ∞ .
□
5.5. The Wegner estimate.
We continue with the celebrated ‘Wegner estimate’. This result due to Wegner
[140] shows not only the regularity of the density of states, it is also a key ingredient
to prove Anderson localization. We set N Λ (E) := N(H Λ , E).
THEOREM 5.23. (Wegner estimate ) Suppose the measure P 0 has a bounded density
g, (i.e. P 0 (A) = ∫ A g(λ)dλ, ||g|| ∞ < ∞) then
E ( N Λ (E + ε) − N Λ (E − ε) ) ≤ C ‖ g ‖ ∞ |Λ| ε . (5.60)
Before we prove this estimate we note two important consequences.
COROLLARY 5.24. Under the assumption of Theorem 5.23 the integrated density
of states is absolutely continuous with a bounded density n(E).
Thus N(E) = ∫ E
−∞
n(λ) dλ. We call n(λ) the density of states. Sometimes, we
also call N the density of states, which, we admit, is an abuse of language.
COROLLARY 5.25. Under the assumptions of Theorem 5.23 we have for any E
and Λ
P ( dist(E, σ(H Λ )) < ε ) ≤ C ‖ g ‖ ∞ ε |Λ| . (5.61)