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

47
Thus from (5.72) we have
Since ∫ +M
−M
∫ +M
−M
So, (5.70) implies
∂
∂V j
tr ( ϱ(H Λ (V Λ ) − η) ) g(V j ) dV j ≤ || g|| ∞ .
g(v)dv = 1 we conclude from (5.71) and (5.72) that
( ∂
E tr ( ϱ (H Λ (V Λ ) − η) )) ≤ || g|| ∞ .
∂V j
E ( N Λ (E + ε) − N Λ (E − ε) ) ≤ 4 || g|| ∞ | Λ| ε . (5.75)
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PROOF (Lemma) : Since B is a positive symmetric rank one operator it is of
the form B = c |h〉〈h| with c ≥ 0, i.e. B ϕ = c 〈h, ϕ〉 h for some h.
By the min-max principle (Theorem 3.1)
Ẽ n = sup
ψ 1 ,...,ψ n−1
≤
≤
≤
sup
ψ 1 ,...,ψ n−1
sup
ψ 1 ,...,ψ n−1
sup
ψ 1 ,...,ψ n−1 ,ψ n
inf
ϕ⊥ψ 1 ,...,ψ n−1
||ϕ||=1
inf
ϕ⊥ψ 1 ,...,ψ n−1
ϕ⊥h ||ϕ||=1
inf
ϕ⊥ψ 1 ,...,ψ n−1 ,h
||ϕ||=1
inf
ϕ⊥ψ 1 ,...,ψ n−1 ,ψn
||ϕ||=1
〈ϕ, A ϕ〉 + c |〈ϕ, h〉| 2
〈ϕ, A ϕ〉 + c |〈ϕ, h〉| 2
〈ϕ, A ϕ〉
〈ϕ, A ϕ〉
= E n+1 . (5.76)
By the Wegner estimate we know that any given energy E is not an eigenvalue of
(H ω ) ΛL for almost all ω. On the other hand it is clear that for any given ω there
are (as a rule |Λ L |) eigenvalues of (H ω ) ΛL .
This simple fact illustrates that we are not allowed to interchange ‘any given E’
and ‘for P−almost all ω’ in assertions like the one above. What goes wrong is that
we are trying to take an uncountable union of sets of measure zero. This union may
have any measure, if it is measurable at all.
In the following we demonstrate a way to overcome these difficulties (in a sense).
This idea is extremely useful when we want to prove pure point spectrum.
THEOREM 5.27. If Λ 1 , Λ 2 are disjoint finite subsets of Z d , then
P ( There is an E ∈ R such that dist(E, σ(H Λ1 )) < ε and dist(σ(E, H Λ2 ))) < ε )
≤ 2 C ‖ g ‖ ∞ ε | Λ 1 || Λ 2 | .
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